Transformation Geometry

Peter McNamara

April 21, 2004

1 Problems

1. Let ABC be a triangle, right angled at A, D the foot of the altitude from A, and E, F the incentres of triangles ABD and ACD respectively. EF intersects AB and AC in K and L. Show AK = AL.

2. Let ABCD be a convex quadrilateral. On the segments AB, BC, CD, DA are chosen points M, N, P, Q respectively, such that

AQ = DP = CN = BM

. Show that if MNP Q is a square, then ABCD is a square.

3. Prove Feuerbach’s theorem; that the nine point circle is tangent to the incircle and the three excircles.

4. In the plane we are given two circles intersecting at X and Y. Prove that there exist four points with the following property: For every circle touching the given circles at A and B, and meeting the line XY at C and D, each of the lines AC, AD, BC, BD passes through one of these four points.

5. Let C be a circle inside another circle Γ. Let T_{0} be a circle tangent externally to C and internally to Γ. For each k ≥ 1, let T_{k} be the circle, distinct from T_{k−2} that is externally tangent to C and T_{k−1} and internally tangent to Γ. Prove that if T_{n} = T_{0} for one particular such circle T_{0}, then T_{n} = T_{0} for all possible positions of T_{0}.

6. Let A_{1} be the centre of the square inscribed in an acute triangle ABC with two vertices on BC and one vertex each on AB and CA. Similarly define B_{1} and C_{1}. Prove that the lines AA_{1}, BB_{1} and CC_{1} are concurrent.

7. (APMO 1998 Q4) Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line passing through D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. Let M and N be the midpoints of the line segments BC and EF respectively. Prove that AN is perpendicular to NM.

8. Two common tangents of two intersecting circles meet at a point A. Let B be a point of intersection of the two circles, and C and D be the points in which one of the tangents touches the circles. Prove that the line AB is tangent to the circle passing through B, C and D.

9. Let ABC be an acute triangle with circumcentre O and circumradius R. Let AO meet the circumcircle of OBC again at D, BO meet the circumcircle of OCA again at E, CO meet the circumcircle of OAB again at F. Show that OD · OE · OF ≥ 8R^{3}.

10. Circle Γ_{1} is internally tangent to circle Γ_{2} at D. At a point B on Γ_{1}, different from D, the tangent to Γ_{1} is drawn and intersects Γ_{2} at A and C. Prove that BD bisects ∠ADC.

11. Let ABC be a triangle, let Γ be its incircle and Γ_{a}, Γ_{b} and Γ_{c} be three circles orthogonal to Γ passing through (B, C), (A, C) and (A, B), respectively. The circles Γ_{a} and Γ_{b} meet again in C′; and in the same way we obtain the points B′ and A′. Prove that the radius of the circumcircle of A′B′C′ is half the radius of Γ.

12. ABCD is a quadrilateral inscribed in the circle with diameter AD. Let O be the midpoint of AD, M the intersection of BC and AD, and K the second intersection point of the circumcircles of triangles BAO and CDO. Prove that ∠MKO = 90^{∘}.

13. (2000 IMO Q6) Let AH_{1}, BH_{2} CH_{3} be the altitudes of an acute angled triangle ABC. The incircle of the triangle ABC touches the sides BC, CA and AB at T_{1}, T_{2} and T_{3} respectivley. Let the lines ℓ_{1}, ℓ_{2} and ℓ_{3} be the reflections of the lines H_{2}H_{3}, H_{3}H_{1} and H_{1}H_{2} in the lines T_{2}T_{3}, T_{3}T_{1} and T_{1}T_{2} respectively. Prove that ℓ_{1}, ℓ_{2} and ℓ_{3} determine a triangle whose vertices lie on the incircle of the triangle ABC.

14. On the sides AB, BC, CD, DA of a quadrilateral ABCD, we construct alternately to the outside and inside, regular triangles with vertices Y, W, X, Z respectively. Show that Y W XZ is a parallelogram.

15. Two circles intersect in points A, B. A line ℓ that contains the point A intersects the circles again in the points C, D. Let M, N be the midpoints of the arcs BC and BD respectively, which do not contain the point A, and let K be the midpoint of the segment CD. Show that ∠MKN = 90^{∘}.