April 2005 1

Plane Geometry (J)

1. Prove that AC is perpendicular to BD if and only if AB^{2}+CD^{2} = BC^{2}+DA^{2}.

2. Let D, E and F be points on sides BC, CA and AB of triangle ABC such that AD, BE and CF are concurrent. Let P be this point of concurrence. Prove that

\frac{AP}{P D} = \frac{AE}{EC} + \frac{AF}{F B} .

3. Let ABC be a triangle and D, E, F the midpoints of BC, CA, AB respectively. Prove that ∠DAC = ∠ABE if and only if ∠AF C = ∠ADB.

4. (Quadrilateral version of Menelaus’ Theorem) Let ABCD be a quadrilateral and suppose that a line ℓ intersects AB at K, BC at L, CD at M and DA at N. Prove that

AK · BL · CM · DN = KB · LC · MD · NA

5. (radical axis theorem - you can quote this) Three circles pairwise intersect. Prove that the three chords common to pairs of circles are concurrent.

6. O is the midpoint of the base BC of an isosceles triangle ABC. A circle is drawn with centre O and tangent to the equal sides AB, AC. P is a point on AB and Q is a point on AC. If P Q is also tangent to this circle, prove that P B · CQ = (\frac{1}{2}BC)^{2}.

7. Suppose the incircle of triangle ABC touches sides BC, CA and AB at points A_{1}, B_{1} and C_{1} respectively. A_{2} is the reflection of A_{1} about the perpendicular bisector of BC. Similarly define B_{2} and C_{2}. Prove that AA_{i}, BB_{i} and CC_{i} are concurrent for i = 1, 2.

8. (Tournament of The Towns, Junior O level, Autumn 1997) Let ABCD be a square and X, Y be points on the sides BC and CD respectively such that AY bisects ∠DAX. Show that AX = BX + DY.

9. Prove that a quadrilateral with side lengths a, b, c and d in that order can have a circle inscribed inside it tangent to all four sides if and only if a + c = b + d.

10. Let ABCD be a quadrilateral. Prove that the incircles of ABC and CDA are tangent to each other if and only if the incircles of BCD and DAB are tangent to each other.

11. (1998 IMO Q1) In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P, where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP have equal areas.

12. (1998 IMO Q5) Let I be the incenter of triangle ABC. Let the incircle of ABC touch the sides BC, CA, and AB at K, L, and M, respectively. The line through B parallel to MK meets the lines LM and LK at R and S, respectively. Prove that angle RIS is acute.