Altshiller-Court Ch 2: Similitude and Homothecy

Other Chapters

1. Geometric Constructions
2. Similitude and Homothecy

Chapter 2: Similitude and Homothecy

A. Similitude

1. If in two triangles two pairs of sides are proportional, and the angle opposite the longer of the two sides considered in one triangle is equal to the corresponding angle in the other triangle, show that the two triangles are similar. Consider the case when the given angle lies opposite the shorter side.

2. If the corresponding sides of two triangles are perpendicular, show that the two triangles are similar.

Construct a triangle, given:

3. $A, B, 2p$.

4. $A, B, b + c$.

5. $A, B, h_a - h_b$.

6. $a:b:c, R$.

7. $A, a:c, h_c$.

8. $A, a:b, 2p$.

9. $B - C, a:(b + c), m_a + m_b$.

10. $a:b, b:c, t_a + t_b - t_c$.

11. Construct a triangle given an angle, the bisector of this angle, and the ratio of the segments into which this bisector divides the opposite side.

12. Construct a right triangle given the perimeter and the ratio of the squares of the two legs.

13. Construct a triangle given the area and the angles which a median makes with the two including sides.

14. Construct a parallelogram given the ratios of one side to the two diagonals, and the area.

15. Given a circle and two radii, produced, draw between them a tangent to the circle which shall be divided by the point of contact in a given ratio.

16. In a given circle to inscribe an isosceles triangle given the sum of its base and altitude.

17. Construct a triangle given $a, A, mb + nc = s$, where $m$ and $n$ are two given constants.

18. If in two triangles two angles are equal and two other angles are supplementary, show that the sides opposite the equal angles are proportional to the sides opposite the supplementary angles.

B. Homothecy

1. Show that a homothecy is determined, given: (a) the homothetic center and a pair of corresponding points; (b) the homothetic ratio and a pair of corresponding points; (c) two pairs of corresponding points.

2. What figure is the homothetic of a parallelogram? Of a rectangle? Of a square?

3. $A, A'$ and $B, B'$ are two pairs of homologous points in two homothetic figures $(F), (F')$, and $M$ is a point of the figure $(F)$. The parallels through $A', B'$ to the lines $AM, BM$, respectively, intersect in $M'$. Prove that $M, M'$ are two corresponding points in the two figures $(F), (F')$.

4. If two triangles are homothetic, show that their circumcenters, etc., are homologous points, and their altitudes, medians, etc., are homologous lines in the two homothetic figures.

5. Through a given point to draw a line passing through the inaccessible point of intersection of two given lines.

6. Construct a triangle given $A, a - b, a - c$.

7. Construct a triangle given $A, a + b, a - c$.

8. Draw a line parallel to the base of a given trapezoid so that the segment intercepted on it by the nonparallel sides of the trapezoid shall be trisected by the diagonals.

9. In a given circle to draw a chord which shall be trisected by two radii given in position.

10. Construct a triangle given a median and the two angles which this median makes with the including sides.

11. Through a given point to draw a line so that the two segments determined on it by three given concurrent lines shall have a given ratio.

12. Given three concurrent lines and a fourth line, to draw a secant so that the three segments determined on it by the four lines shall have given ratios.

In a given triangle to inscribe:

13. A triangle whose sides shall be parallel to the internal bisectors of the given triangle. Two solutions.

14. A triangle whose sides shall be perpendicular to the sides of the given triangle. Two solutions.

15. A square.

16. A rectangle similar to a given rectangle.

17. A parallelogram given the ratio of its sides and the angle between the diagonals.

18. Given a triangle $ABC$, construct a square so that two vertices shall lie on $BA$ and $CA$, both produced, and the other two vertices on the side $BC$.

19. In a given semicircle to inscribe a rectangle similar to a given rectangle.

20. In a given circular sector to inscribe a square. Two cases: (i) one vertex or (ii) two vertices lie on the circumference.

21. Construct a square so that two vertices shall lie on a given line and the other two on a given circle.

22. Draw a parallel to the base of a given triangle so that the segment intercepted on it by the other two sides shall subtend a given angle at a given point of the base.

23. If $m$ is the side of the square inscribed in the triangle $ABC$ so that two vertices lie on the side $BC = a$, and $h$ is the altitude upon $BC$, show that $m(a + h) = ah$.

24. Through a given point on a circle to draw a chord so that it shall be bisected by a given chord.

25. Through a given point to draw a secant so that the segment intercepted on it by a given line and a given circle shall be divided by the given point in a given ratio.

26. Through a given point to draw a secant so that the ratio of the distances of the given point from the points of intersection of the secant with a given circle shall have a given value.

27. On two given circles to find two points collinear with, and equidistant from, a given point.

28. Construct a triangle given two sides and the bisector of the included angle $(b, c, t_a)$.

29. Draw a line on which two concentric circles shall determine two chords having a given ratio.

30. Given three concentric circles to draw a secant so that the segment between the first and the second circles shall be equal to the segment between the second and the third circles.

31. A variable point $P$ moves on a fixed circle, center $C$, and $A$ is a fixed point. Find the locus of the point of intersection of the line $AP$ with the internal bisector of the angle $ACP$.

32. A variable triangle has a fixed base and a fixed circumcircle (i.e., circumscribed circle). Find the locus of the midpoints of the lateral sides, and the locus of the midpoint of the segment joining the midpoints of the lateral sides.

33. From the given points $B$, $C$ the perpendiculars $BB'$, $CC'$ are dropped upon a variable line $AB'C'$ passing through a fixed point $A$ collinear with $B$ and $C$. Find the locus of the point $M = (BC', B'C)$.

34. On the base $BC$ of a given triangle $ABC$ find a point $P$ such that $AP^2:BP \cdot PC$ shall have a given value.

35. Construct a triangle similar to a given triangle so that one vertex shall coincide with a given point and the other two vertices shall lie on two given lines.

36. In a given triangle to inscribe a triangle similar to a (second) given triangle, one of the vertices being given.

37. Construct a triangle similar to a given triangle so that its vertices shall lie on three given lines.

38. The vertex $A$ of the variable triangle $APQ$ is fixed, and $P$ moves on a fixed line $CD$; $AP$ meets a fixed line parallel to $CD$ in the point $R$, and $PQ = AR$; the angle $APQ$ is constant. Prove that the locus of $Q$ is a straight line.

39. A variable regular hexagon has a fixed vertex and its center describes a straight line. Show that the remaining vertices describe straight lines and that these lines are concurrent.

40. Construct a triangle similar to a given triangle so that one vertex shall coincide with a given point and the other two vertices shall lie on two given circles.

41. Construct a triangle similar to a given triangle so that its vertices shall lie on three given circles.

42. Construct a triangle similar to a given triangle so that one vertex shall coincide with a given point, another shall lie on a given circle, and the third on a given line.

Supplementary Exercises

1. The lines $AL$, $BL$, $CL$ joining the vertices of a triangle $ABC$ to a point $L$ meet the respectively opposite sides in $A'$, $B'$, $C'$. The parallels through $A'$ to $BB'$, $CC'$ meet $AC$, $AB$ in $P$, $Q$, and the parallels through $A'$ to $AC$, $AB$ meet $BB'$, $CC'$ in $R$, $S$. Show that the four points $P$, $Q$, $R$, $S$ are collinear.

2. $ABCD$ is a rhombus, and $P, Q, R, S$ are the circumcenters of the triangles $BCD$, $CDA$, $DAB$, $ABC$. Prove that the midpoints of the segments $AP, BQ, CR, DS$ form a rhombus similar to $ABCD$.

3. Construct a triangle $ABC$ given its circumcircle, center $O$, so that area $OBC:OCA:OAB = p:q:r$, where $p, q, r$ are given line segments.

4. Locate two points $D$, $E$ on the sides $AB$, $AC$ of a triangle $ABC$ such that $AD:DE:EC = p:q:r$, where $p, q, r$ are given line segments.