Selected Geometry Olympiad Problems II Russelle Guadalupe May 14, 2011
Problems 1. [Bulgaria1997] Let ABCD be a convex quadrilateral such that ∠DAB = ∠ABC = ∠BCD. Let H and O be the orthocenter and circumcenter of triangle ABC. Prove that D, O, H are collinear. 2. [Switz2011] Let ABC a triangle with ∠CAB = 90◦ and L a point on the segment BC. The circumcircle of triangle ABL intersects AC at M and the circumcircle of triangle CAL intersects AB at N . Show that L, M and N are collinear. 3. Circle O has diameter AB and CD is a chord. AD ∩ BC = K . Point E lies on CD such that KE ⊥ AB. Lines EA, EB cut OC, OD at F, G, respectively. Prove that F, K, G are collinear on a line parallel to AB. 4. Let Γ be the circumcircle of 4ABC with center O. E is the excenter opposite A. Draw a line l through E perpendicular to AE. Let X, Y be points on l such that ∠XAO = ∠Y AO = ∠BAE. Prove that the incenter of 4AXY lies on Γ. 5. [China1997] Let ABCD be a cyclic quadrilateral. The lines AB and CD meet at P , the lines AD and BC meet at Q. Let E and F be the points of tangency of Q to the circumcircle of ABCD. Prove that P, E, F are collinear. 6. [Korea1997] In an acute triangle ABC with AB 6= AC, the angle bisector of A meets BC at V . D is the foot of the altitude from A to BC. If the cirucmcircle of AV D meets CA and AB at E and F respectively, prove that AD, BE, CF are concurrent. 7. [Moldova2007] Let M, N be points inside the angle ∠BAC such that ∠M AB = ∠N AC. If M1 , M2 and N1 , N2 are the projections of M and N on AB, AC respectively, prove that M, N and P = M1 N2 ∩ N1 M2 are collinear. 8. [SL2009 G4] Given a cyclic quadrilateral ABCD, let the diagonals AC and BD meet at E and the lines AD and BC meet at F . The midpoints of AB and CD are G and H, respectively. Show that EF is tangent at E to the circle through the points E, G and H.
More Problems 1. If H is the orthocenter of 4ABC, prove that HA · HB · AB + HB · HC · BC + HC · HA · AC = AB · BC · CA 2. [Iran TST 2005] Assume that 4ABC is isosceles with AB = AC. Suppose that P is a point in the extension of side BC. Points X and Y are points on AB and AC such that P X k AC and P Y k AB. Let T be the midpoint of arc AC not containing A. Prove that P T ⊥ XY . 3. [SL2009 G4] Given a cyclic quadrilateral ABCD, let the diagonals AC and BD meet at E and the lines AD and BC meet at F . The midpoints of AB and CD are G and H, respectively. Show that EF is tangent at E to the circle through the points E, G and H. 4. Cyclic quadrilateral ABCD is inscribed in a circle ω centered at O. The tangents to ω at B and C meet the line AD at M, N respectively. Let E = BN ∩ CM and F = AE ∩ BC. If L is the midpoint of BC, show that the circumcircle of 4DLF is tangent to ω at D. 5. In triangle ABC with orthocenter H, A0 is the foot of the altitude from A and D is a point on AB. Suppose that line DH meets the circumcircle of triangle ADA0 again at E. Prove that CE ⊥ ED. 6. [CXM1 2939] In triangle ABC with incenter I, BI ∩ AC = D and CI ∩AB = E. The bisector2 of ∠BIC meets BC, DE at P, Q respectively. Suppose that P I = 2QI. Prove that ∠A = 60◦ . 7. [CXM 2940] In triangle ABC, D, E lie on sides AC, AB respectively such that BD bisects ∠B and CE bisects ∠C. If ∠ADE − ∠AED = 60◦ , prove that ∠C = 120◦ . 8. [CXM 2876] In triangle ABC, I is the incenter, O is the circumcenter and M is the midpoint of BC. Suppose that IO ⊥ AM . Prove that 1 1 2 = + . BC AB AC 9. [CXM 3015] In 4ABC with incenter I, BC < AB, and BC < AC. The exterior bisectors of ∠B and ∠C meet AC, AB at D, E respectively. Prove that BD (AI 2 − BI 2 )CI . = CE (AI 2 − CI 2 )BI
10. [CXM 2936] In 4ABC, ∠B = 2∠C and ∠A > 90◦ . The perpendicular to AC through C meets AB at D. Prove that 1 1 2 − = . AB BD BC 1 Crux
Mathematicorum, (Note: Problems 6 to 10 are proposed by Professor Toshio Seimiya.) 2 This term will mean the internal bisector, unless otherwise stated.