Power of a Point Problems

Introduction

Here’s another entry in our themed problem series! Like our algebraic equations with square roots collection, this page gathers competition problems around a single powerful concept: the Power of a Point theorem.

AMC/AIME Problems

Power of a Point problems are relatively rare in AMC and AIME competitions. When they do appear, they tend to be high-numbered questions (AMC10 #23, AMC12 #16, #17, #24, #24, AIME #10, #15, #15), making them quite challenging on the AMC/AIME scale.

How rare? Searching "amc 10" "power of a point" site:artofproblemsolving.com on Google returns about 7 problems that reference Power of a Point somewhere in their solution. For AMC 12, a similar search finds around 14 problems. While I don’t normally reference AoPS solutions on this blog, these searches suggest the theorem appears more often than I initially thought.

However, you can’t always trust AoPS to be the definitive source for all solutions. To illustrate, I spent a little bit of time trying to come up with brand new Power of a Point solutions to AMC 10 problems. It didn’t take me that long to identify the following problem:

2001 AMC 10 #24

In trapezoid $A B C D$ we have $\overline{A B}$ and $\overline{C D}$ perpendicular to $\overline{A D}$ with $A B+C D=B C$, $A B<C D$, and $A D=7$. What is $A B \cdot C D$ ?

(A) 12 (B) 12.25 (C) 12.5 (D) 12.75 (E) 13

Solution

All 5 solutions on AoPS appear to actually be the same solution (a common problem with the wiki) using the Pythagorean Theorem. Below is my weird solution involving Power of a Point. Is it the best solution? Probably not. Is it the most beautiful? You decide!

First, our initial diagram:

Extend line $CD$ past $D$ (to the left) to point $B'$ such that $DB' = AB$. Now $\triangle BCB'$ is isosceles because $AB+CD=BC$ implies $CB' = CD + DB' = CD + AB = BC$. So $\triangle BCB'$ is isosceles with $BC = CB'$. In addition, the line segments $BB'$ and $AD$ bisect each other at midpoint, $M$. Since $\triangle BCB'$ is isosceles, the altitude from $C$ to base $\overline{BB'}$ passes through $M$ and is perpendicular to $\overline{BB'}$. This all sounds more complicated than it is - here’s our construction so far:

Now for the magic! Draw the circumcircle for $\triangle B'MC$, noting that the circumcenter is on the midpoint of $B'C$ since $\angle B'MC=90^\circ$:

Remembering that a radius perpendicular to a chord bisect the chord, we see that $MD=DM'$. As is the case with most synthetic geometry solutions, we’re now essentially at the solution without having done any arithmetic or algebra whatsoever! Power of a Point on $D$ immediately yields the answer:

$$\begin{align*} DB' \cdot CD &= DM \cdot DM'\\ AB \cdot CD &= DM^2\\ AB \cdot CD &= 3.5^2 = 12.25 \end{align*}$$

Additional Problems

Below is my list of Power of a Point problems from AMC/AIME. AoPS may have a more comprehensive collection, so feel free to check them out as well.

1. 2013 AMC 10A #23 / 12A #19: In $\triangle A B C, A B=86$, and $A C=97$. A circle with center $A$ and radius $A B$ intersects $\overline{B C}$ at points $B$ and $X$. Moreover $\overline{B X}$ and $\overline{C X}$ have integer lengths. What is $B C$?
   
   (A) 11 (B) 28 (C) 33 (D) 61 (E) 72

2. 2012 AMC 12A #16: Circle $C_{1}$ has its center $O$ lying on circle $C_{2}$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_{1}$ lies on circle $C_{2}$ and $X Z=13, O Z=11$, and $Y Z=7$. What is the radius of circle $C_{1}$?
   
   (A) 5 (B) $\sqrt{26}$ (C) $3 \sqrt{3}$ (D) $2 \sqrt{7}$ (E) $\sqrt{30}$

3. 2006 AMC 12A #17: Square $A B C D$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{B E}$. Point $F$ lies on the circle, on the same side of $\overline{B E}$ as $A$. Segment $A F$ is tangent to the circle, and $A F=\sqrt{9+5 \sqrt{2}}$. What is $r / s$?

(A) $\frac{1}{2}$ (B) $\frac{5}{9}$ (C) $\frac{3}{5}$ (D) $\frac{5}{3}$ (E) $\frac{9}{5}$

4. 2008 AMC 12A #24: Triangle $A B C$ has $\angle C=60^{\circ}$ and $B C=4$. Point $D$ is the midpoint of $\overline{B C}$. What is the largest possible value of $\tan (\angle B A D)$?
   
   (A) $\frac{\sqrt{3}}{6}$ (B) $\frac{\sqrt{3}}{3}$ (C) $\frac{\sqrt{3}}{2 \sqrt{2}}$ (D) $\frac{\sqrt{3}}{4 \sqrt{2}-3}$ (E) 1

5. 2000 AMC 12 #24: If circular arcs $\overset{\frown}{AC}$ and $\overset{\frown}{BC}$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\overset{\frown}{AC}$ and $\overset{\frown}{BC}$, and to $\overline{A B}$. If the length of $\overset{\frown}{BC}$ is 12 , then the circumference of the circle is

(A) 24 (B) 25 (C) 26 (D) 27 (E) 28

6. 2017 AMC 12A #24: Quadrilateral $A B C D$ is inscribed in circle $O$ and has sides $A B=3$, $B C=2, C D=6$, and $D A=8$. Let $X$ and $Y$ be points on $\overline{B D}$ such that
   

   $\frac{D X}{B D}=\frac{1}{4} \quad \text { and } \quad \frac{B Y}{B D}=\frac{11}{36}$ Let $E$ be the intersection of line $A X$ and the line through $Y$ parallel to $\overline{A D}$. Let $F$ be the intersection of line $C X$ and the line through $E$ parallel to $\overline{A C}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $C X$. What is $X F \cdot X G$?
   
   (A) 17 (B) $\frac{59-5 \sqrt{2}}{3}$ (C) $\frac{91-12 \sqrt{3}}{4}$ (D) $\frac{67-10 \sqrt{2}}{3}$ (E) 18

7. 2016 AIME II #10: Triangle $A B C$ is inscribed in circle $\omega$. Points $P$ and $Q$ are on side $\overline{A B}$ with $A P<A Q$. Rays $C P$ and $C Q$ meet $\omega$ again at $S$ and $T$ (other than $C$ ), respectively. If $A P=4, P Q=3, Q B=6, B T=5$, and $A S=7$, then $S T=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

8. 2016 AIME I #15: Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $X$ and $Y$. Line $\ell$ is tangent to $\omega_{1}$ and $\omega_{2}$ at $A$ and $B$, respectively, with line $A B$ closer to point $X$ than to $Y$. Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_{1}$ again at $D \neq A$ and intersecting $\omega_{2}$ again at $C \neq B$. The three points $C, Y$, and $D$ are collinear, $X C=67, X Y=47$, and $X D=37$. Find $A B^{2}$.

9. 2015 AIME II #15: Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii 1 and 4 , respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $B C$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle D B A$ and $\triangle A C E$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

10. 2019 AIME I #15: Let $\overline{A B}$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $\overline{A B}$. Circle $\omega_{1}$ passes through $A$ and $P$ and is internally tangent to $\omega$. Circle $\omega_{2}$ passes through $B$ and $P$ and is internally tangent to $\omega$. Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $P$ and $Q$. Line $P Q$ intersects $\omega$ at $X$ and $Y$. Assume that $A P=5, P B=3, X Y=11$, and $P Q^{2}=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Other Competition Problems

Here are Power of a Point problems from some other famous competitions:

1. 1997 ARML Team #2: Equilateral triangle $A B C$ is inscribed in circle $O$. Let $D$ and $E$ be midpoints of $\overline{A C}$ and $\overline{A B}$. If $\frac{D E}{E F}$ can be written as $\frac{a+\sqrt{b}}{c}$, for integers $a, b$, and $c$ in simplest form, compute the ordered triple $(a, b, c)$.

2. 2022 NYCIML Spring Senior A #4: Let $\triangle A B C$ have circumradius 8 , and let $P$ be a point on the circumcircle. If the distances from $P$ to the lines $A B, A C$, and $B C$ are 1, 1, and 9, respectively, compute $B C$.

3. 2020 NYCIML Spring Senior A #4: In $\triangle A B C$, the $A$-angle bisector intersects $B C$ at $D$, and $E$ is the reflection of $A$ over $D$. Suppose that $A, B, C$, and $E$ lie on a circle. If $A B=10$ and $A C=16$, compute $B C$.

4. 2019 NYCIML Spring Senior A #4: A circle whose center is in the first quadrant contains the points $(1,2)$ and $(3,6)$ and is tangent to the $x$-axis. The radius of the circle can be expressed in the form $a-\frac{\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers and $b$ is square-free. Find $(a, b, c)$.

5. 2012 JHMT Geometry #4: Circle $O$ has radius 18 . From diameter $A B$, there exists a point $C$ such that $B C$ is tangent to $O$ and $A C$ intersects $O$ at a point $D$, with $A D=24$. What is the length of $B C$?

6. 2019 MAΘ Hustle Geometry #4: Points $T, S, H, I$ lie on a circle. Chords $T H$ and $S I$ are perpendicular and intersect at point $B$. Let $T B=3, I B=2$, and $S B=6$. Compute $H B$.

7. 2021 BMT Geometry #5: Let circles $\omega_{1}$ and $\omega_{2}$ intersect at $P$ and $Q$. Let the line externally tangent to both circles that is closer to $Q$ touch $\omega_{1}$ at $A$ and $\omega_{2}$ at $B$. Let point $T$ lie on segment $\overline{P Q}$ such that $\angle A T B=90^{\circ}$. Given that $A T=6, B T=8$, and $P T=4$, compute $P Q$.

8. 2012 OMO Fall #5: Two circles have radii 5 and 26 . The smaller circle passes through center of the larger one. What is the difference between the lengths of the longest and shortest chords of the larger circle that are tangent to the smaller circle?

9. 2023 BMT Geometry #6: In triangle $\triangle A B C$, let $M$ be the midpoint of $\overline{A C}$. Extend $\overline{B M}$ such that it intersects the circumcircle of $\triangle A B C$ at a point $X$ not equal to $B$. Let $O$ be the center of the circumcircle of $\triangle A B C$. Given that $B M=4 M X$ and $\angle A B C=45^{\circ}$, compute $\sin (\angle B O X)$.

10. 2010 NYCIML Spring Junior #6: Two circles intersect at $A$ and $B, A \neq B$. A line is tangent to one circle at $P$ and the other at $Q$. Line $A B$ intersects $\overline{P Q}$ at $T$. If the radii of the circles are 3 and 5 , and the distance between their centers is 6 , compute $P T$.

11. 2001 MAΘ Proofs #6: Given segments with length $1, a$, and $b$, find a method to construct a segment of length $a b$ using only a straightedge and compass, and prove that it works. (10 points)

12. 2025 MAΘ Theta #6: A sphere of radius 2 is inscribed in a right circular cone whose height is 12 . Find the radius of the cone.
    
    A. $\sqrt{6}$ B. 2.4 C. 3 D. 6 E. NOTA

13. 2024 BMT Geometry #7: In parallelogram $A B C D, E$ is a point on $\overline{A D}$ such that $\overline{C E} \perp \overline{A D}, F$ is a point on $\overline{C D}$ such that $\overline{A F} \perp \overline{C D}$, and $G$ is a point on $\overline{B C}$ such that $\overline{A G} \perp \overline{B C}$. Let $H$ be a point on $\overline{G F}$ such that $\overline{A H} \perp \overline{G F}$, and let $J$ be the intersection of $\overleftrightarrow{E F}$ and $\overleftrightarrow{B C}$. Given that $A H=8, A E=6$, and $E F=4$, compute $C J$.

14. 2009 NYCIML Fall Junior #7: In a circle, chords $\overline{A B}$ and $\overline{C D}$ intersect at $E$, $A E$ is one less than $C E$, $D E$ is one more than $C E$, and $B E$ is twice $C E$. Find $A B$.

15. 2005 ARML Team #7: In the diagram, circle $O$ has a radius of 10 , circle $P$ is internally tangent to $O$ and has a radius of 4. $\overline{Q T}$ is tangent to circle $P$ at $T$ and, if drawn, line $\overleftrightarrow{P T}$ intersects circle $O$ at points $A$ and $B$. Compute the product $T A \cdot T B$.

16. 2023 BMT Geometry #8: A circle intersects equilateral triangle $\triangle X Y Z$ at $A, B, C, D, E$, and $F$ such that points $X, A, B$, $Y, C, D, Z, E$, and $F$ lie on the equilateral triangle in that order. If $A C^{2}+C E^{2}+E A^{2}=1900$ and $B D^{2}+D F^{2}+F B^{2}=2092$, compute the positive difference between the areas of triangles $\triangle A C E$ and $\triangle B D F$.

17. 2021 BMT Geometry #8: Let $\triangle A B C$ be a triangle with $A B=15, A C=13, B C=14$, and circumcenter $O$. Let $\ell$ be the line through $A$ perpendicular to segment $\overline{B C}$. Let the circumcircle of $\triangle A O B$ and the circumcircle of $\triangle A O C$ intersect $\ell$ at points $X$ and $Y$ (other than $A$ ), respectively. Compute the length of $\overline{X Y}$.

18. 2021 JHMMC Grade 8 Round 2 #8: Marj and Nick decide to go swimming in a perfectly circular lake. Marj can swim at 4 miles per hour while Nick can swim at 3 miles per hour. They start at different points on the circumference of the lake and swim in a straight line, stopping when they reach the shore. They only cross paths once. When they cross paths, Marj has been swimming for 25 minutes and Nick has been swimming for 10 minutes. If Marj ends up swimming $\frac{3}{4}$ of a mile more than Nick, the distance Marj swam can be represented as the simplified fraction $\frac{a}{b}$. What is $a+b$?

19. 2019 BMT Geometry #8: Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Let $G$ denote the centroid of $A B C$, and let $G_{A}$ denote the image of $G$ under a reflection across $B C$, with $G_{B}$ the image of $G$ under a reflection across $A C$, and $G_{C}$ the image of $G$ under a reflection across $A B$. Let $O_{G}$ be the circumcenter of $G_{A} G_{B} G_{C}$ and let $X$ be the intersection of $A O_{G}$ with $B C$ and $Y$ denote the intersections of $A G$ with $B C$. Compute $X Y$.

20. 2012 JHMT General #8: Circle $O$ has radius 18 . From diameter $A B$, there exists a point $C$ such that $B C$ is tangent to $O$ and $A C$ intersects $O$ at a point $D$, with $A D=24$. What is the length of $B C$?

21. 2024 BMT Geometry #9: Let $\triangle A B C$ be a triangle with incenter $I$, and let $M$ be the midpoint of $\overline{B C}$. Line $\overleftrightarrow{A M}$ intersects the circumcircle of triangle $\triangle I B C$ at points $P$ and $Q$. Suppose that $A P=13, A Q=83$, and $B C=56$. Find the perimeter of $\triangle A B C$.

22. 2010 JHMT Grabbag #9: Let $\triangle A B C$, be acute with perimeter 100. Let $D$ be a point on $\overline{B C}$. The circumcircles of $A B D$ and $A D C$ intersect $\overline{A C}$ and $\overline{A B}$ at $E$ and $F$ respectively such that $D E=14$ and $D F=11$. If $\angle E B C \cong \angle B C F$, find $\frac{A E}{A F}$.

23. 2022 BMT Geometry #10: In triangle $\triangle A B C, E$ and $F$ are the feet of the altitudes from $B$ to $\overline{A C}$ and $C$ to $\overline{A B}$, respectively. Line $\overleftrightarrow{B C}$ and the line through $A$ tangent to the circumcircle of $A B C$ intersect at $X$. Let $Y$ be the intersection of line $\overleftrightarrow{E F}$ and the line through $A$ parallel to $\overline{B C}$. If $X B=4, B C=8$, and $E F=4 \sqrt{3}$, compute $X Y$.

24. 2023 MAΘ Theta Equations And Inequalities #10: Two secant lines are drawn from point $C$ to the same circle. One line intersects the circle at points $B$ and $A$. The other intersects the circle at points $D$ and $E$. If $C B=x+3$, $C A=2 x+4$, $C D=2 x$, and $C E=x+8$, find $x$.
    
    A. 2 B. 3 C. 4 D. 6 E. NOTA

25. 2015 MAΘ Theta Geometry #11: Find the value of $x$ in the following diagram. ( $\overline{A B}$ is a tangent to the circle.)

A. 11 B. 9 C. $\frac{17}{2}$ D. $\frac{21}{2}$ E. NOTA

26. 2021 NYCIML Fall Senior B #12: Triangle $A B C$ with area 33 is inscribed in a circle. Point $E$ is on the arc between $A C$ which does not contain $B$ and point $D$ is chosen on the arc between $B C$ which does not contain $A$. Let $F$ be the intersection of $A C$ and $E D$, and $G$ be the intersection of $B C$ and $E D$. If $E F=F G=G D$, $A F=6$, $F C=3$, and $C G=2$, compute the area of $B E D$.

27. 2023 BmMT Relay #13: Let $N_{10}$ be the answer to question 10, $N_{11}$ be the sum of the digits of the answer to question 11, and $N_{12}$ be the sum of the digits of the answer to question 12. Two circles $O_{1}$ and $O_{2}$ are externally tangent to line $\ell$ at the same point $A$, and the two circles are externally tangent to each other. Point $B$ lies on line $\ell$ such that $A B=N_{11}$. There is a line through point $B$ that intersects circle $O_{1}$ at points $D$ and $E$, where $B D<B E$ and $B D=N_{10}$, and another line through point $B$ that intersects circle $O_{2}$ at points $F$ and $G$, where $B F<B G$ and $B F=N_{12}$. Compute $\frac{D G}{E F}$.

28. 2021 JHMMC Grade 8 Round 2 #13: Let $\Omega$ be a semicircle with diameter $A B=10$. There exist points $C$ and $D$ on $\Omega$ such that $A C$ and $B D$ intersect at point $E$ inside the semicircle, with $A E=6$ and $C E=2$. If $A D^{2}$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.

29. 2023 NYCIML Fall Junior #15: Rectangle $A B C D$ with $A B=3$ and $B C=4$ is inscribed in a circle. The perpendicular bisector of $\overline{B C}$ intersects the circle at $X$ and $Y$ and $\overline{B C}$ at $M$. Given that $M X<M Y$, compute $M X$.

30. 2018 BMT Team #15: Let triangle $A B C$ have side lengths $A B=13, B C=14, A C=15$. Let $I$ be the incenter of $A B C$. The circle centered at $A$ of radius $A I$ intersects the circumcircle of $A B C$ at $H$ and $J$. Let $L$ be a point that lies on both the incircle of $A B C$ and line $H J$. If the minimal possible value of $A L$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.

31. 2009 NYCIML Fall Junior #16: Two circles intersect at $P$ and $Q$. Point $K$ is on $\overline{P Q}$, and a line through $K$ intersects one circle at $A$ and $B$ and the other circle at $C$ and $D$, and the points are in the order $\overline{\mathrm{ACKBD}}$. If $A C=6, C K=1$, and $K B=2$, find $B D$.

32. 2015 OMO Spring #17: Let $A, B, M, C, D$ be distinct points on a line such that $A B=B M=M C=C D=6$. Circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ and radius 4 and 9 are tangent to line $A D$ at $A$ and $D$ respectively such that $O_{1}, O_{2}$ lie on the same side of line $A D$. Let $P$ be the point such that $P B \perp O_{1} M$ and $P C \perp O_{2} M$. Determine the value of $P O_{2}^{2}-P O_{1}^{2}$.

33. 2014 OMO Fall #17: Let $A B C$ be a triangle with area 5 and $B C=10$. Let $E$ and $F$ be the midpoints of sides $A C$ and $A B$ respectively, and let $B E$ and $C F$ intersect at $G$. Suppose that quadrilateral $A E G F$ can be inscribed in a circle. Determine the value of $A B^{2}+A C^{2}$.

34. 2024 BmMT Relay #18: Let $N_{15}$ be the answer to Problem 15, $N_{16}$ be the answer to Problem 16, and $N_{17}$ be the answer to Problem 17. Triangle $\triangle X Y Z$ has a right angle at $Y$. Points $A$ and $D$ lie on $\overline{X Y}$ and $\overline{Y Z}$, respectively, such that $\overline{A D}$ is parallel to $\overline{X Z}$. Let $\overline{A D}$ intersect the inscribed circle of $\triangle X Y Z$ at points $B$ and $C$, with $A$ closer to $B$ than $C$. Suppose $A B=N_{16}, B C=N_{17}$, and $C D=N_{15}$. Compute the smallest possible value of $X Z$.

35. 2023 NYCIML Spring SophFrosh #18: $\triangle A B C$ is scalene and inscribed in circle $O$. There exists a point $P$ inside $\triangle A B C$ such that line $\overleftrightarrow{C P}$ intersects $\overline{A B}$ at $D$ and circle $O$ again at $E$. $C P=6$, $P D=3$, $D E=2$, and $A D=3$. Given that the area of $\triangle A E D$ is 4, find the area of $\triangle A B C$.

36. 2017 MAΘ Combinatorics And Probability #18: A circle of radius 1 is internally tangent to a circle with radius 2 . A chord of the larger circle is selected at random. If the chord is tangent to the smaller circle, what is the probability that its length is no more than $\sqrt{15}$?
    
    A. $\frac{1}{4}$ B. $\frac{1}{2}$ C. $\frac{3}{4}$ D. 1 E. NOTA

37. 2021 MAΘ Theta Circles And Polygons #21: A circle is drawn, and a tangent and a secant to the circle are drawn from point A, which is outside of the circle. If the tangent line intersects the circle at point $B$, and the secant line intersects the circle at points C and D, with $A C<A D$, then what is the length of CD, given that $A B=12$ and $A C=9$?
    
    A. 3 B. 16 C. 7 D. 25 E. NOTA

38. 2020 BMT Team #21: Let $\triangle A B C$ be a right triangle with legs $A B=6$ and $A C=8$. Let $I$ be the incenter of $\triangle A B C$ and $X$ be the other intersection of $A I$ with the circumcircle of $\triangle A B C$. Find $\overline{A I} \cdot \overline{I X}$.

39. 2023 MAΘ Theta Circles And Polygons #22: A circle has chords MD and PN that intersect at point U . If $M U=15, P U=10$, and $U N=U D+7$, find the length of MD.
    
    A. 35 B. 26 C. 31 D. 29 E. NOTA

40. 2025 GiM Guts #23: Points $A, B$, and $C$ exist on a parabola with focus $F$, directrix $\ell$, and vertex $B$. Let $D$ be the foot of the altitude from $F$ to $A B$ and $E$ the foot of the altitude from $F$ to $B C$. Denote the intersection between $\overline{F D}$ and $\ell$ as $X$ and the intersection between $\overline{F E}$ and $\ell$ as $Y$. If $F A=33, F B=26, F C=46, A B=25$, and $B C=52$, find the area of $\triangle F X Y$.

41. 2022 NYCIML Spring Senior B #23: Square $A B C D$ has sidelength 8, and circles of radius 8 are centered at $C$ and $D$, and the circles intersect at point $E$ inside the square. Find the area of $\triangle E C D$.

42. 2017 JHMMC Grade 8 #24: Suppose you have a circle with center $O$ and diameter 15 . Point $P$ is outside the circle and $A$ is a point such that $P A$ is a tangent. Extend $A$ through $O$ to get $C$ on the circle and let $B$ be the intersection of $P C$ and the circle. $P B=16$. Find $P C$.

43. 2023 MAΘ Theta Area And Volume #26: In square ABCD of side length 20, a circle passes through $\mathrm{A}$, $\mathrm{B}$ and the midpoint of $\overline{C D}$. What is the area of the circle?
    
    A. $144 \pi$ B. $225 \pi$ C. $100 \pi$ D. $\frac{625}{4} \pi$ E. NOTA

44. 2015 OMO Spring #27: Let $A B C D$ be a quadrilateral satisfying $\angle B C D=\angle C D A$. Suppose rays $A D$ and $B C$ meet at $E$, and let $\Gamma$ be the circumcircle of $A B E$. Let $\Gamma_{1}$ be a circle tangent to ray $C D$ past $D$ at $W$, segment $A D$ at $X$, and internally tangent to $\Gamma$. Similarly, let $\Gamma_{2}$ be a circle tangent to ray $D C$ past $C$ at $Y$, segment $B C$ at $Z$, and internally tangent to $\Gamma$. Let $P$ be the intersection of $W X$ and $Y Z$, and suppose $P$ lies on $\Gamma$. If $F$ is the $E$-excenter of triangle $A B E$, and $A B=544, A E=2197, B E=2299$, then find $m+n$, where $F P=\frac{m}{n}$ with $m, n$ relatively prime positive integers.

45. 1992 AHSME #27: A circle of radius $r$ has chords $\overline{A B}$ of length 10 and $\overline{C D}$ of length 7 . When $\overline{A B}$ and $\overline{C D}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle A P D=60^{\circ}$ and $B P=8$, then $r^{2}=$
    
    (A) 70 (B) 71 (C) 72 (D) 73 (E) 74

46. 2022 NYCIML Spring Senior A #28: A circle has points $A, B, C, D$, and $E$ on it, in that order, and $\overline{B E}$ intersects $\overline{A C}$ and $\overline{A D}$ at points $X$ and $Y$ respectively. $X$ bisects $\overline{A C}, Y$ bisects $\overline{A D}$, and $X$ and $Y$ trisect $\overline{B E}$. If $\overline{A B}=12$, find the length of $\overline{A C}$.

47. 2017 OMO Spring #29: Let $A B C$ be a triangle with $A B=2 \sqrt{6}, B C=5, C A=\sqrt{26}$, midpoint $M$ of $B C$, circumcircle $\Omega$, and orthocenter $H$. Let $B H$ intersect $A C$ at $E$ and $C H$ intersect $A B$ at $F$. Let $R$ be the midpoint of $E F$ and let $N$ be the midpoint of $A H$. Let $A R$ intersect the circumcircle of $A H M$ again at $L$. Let the circumcircle of $A N L$ intersect $\Omega$ and the circumcircle of $B N C$ at $J$ and $O$, respectively. Let circles $A H M$ and $J M O$ intersect again at $U$, and let $A U$ intersect the circumcircle of $A H C$ again at $V \neq A$. The square of the length of $C V$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100 m+n$.

48. 2022 NYCIML Spring Senior B #30: A circle has points $A, B, C, D$, and $E$ on it, in that order, and $\overline{B E}$ intersects $\overline{A C}$ and $\overline{A D}$ at points $X$ and $Y$ respectively. $X$ bisects $\overline{A C}, Y$ bisects $\overline{A D}$, and $X$ and $Y$ trisect $\overline{B E}$. If $\overline{A B}=12$, find the length of $\overline{A C}$.

49. 2023 JHMMC Grade 7 #34: A sphere of radius 1 is inscribed in a cube, and a line segment connects one point where a face of the cube is tangent to the sphere to a vertex on the opposite side of the cube. If $\frac{a}{b}$ of the segment’s total length lies outside the sphere, where $a$ and $b$ are relatively prime positive integers, compute $a+b$.

50. 2020 JHMMC Grade 8 Round 1 #35: Consider triangle $A B C$ with circumcenter $O$, such that $A O$ is parallel to $B C$. If the circumradius of $A B C$ has length 4 and side $B C$ has length 5 , then compute the value of $A C^{2}$.

51. 2013 JHMMC Grade 5 #36: In Circle $O$, chord $\overline{A B}$ is bisected by chord $\overline{C D}$ at $E$. If $C D=10$ and $D E=2$, compute the length of $A B$.

52. 2023 JHMMC Grade 8 #38: A square is inscribed in a circle. Chord $\overline{A B}$ of the circle intersects the square at points $X$ and $Y$. If $A X=X Y=Y B=6$, what is the area of the square?

53. 2001 MAΘ Mu #49: $A B$ and $C D$ are perpendicular diameters of circle $O$. $C M$ is a chord that intersects $A B$ at $E$. If $C E=4$ and $E M=3$, what is the area of $O$?
    
    (A) $14 \pi$ (B) $15 \pi$ (C) $16 \pi$ (D) $17 \pi$ (E) NOTA