Lotus Ratio Lemma P Set [PDF]

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piyush kumar jha — March 18, 2023



Problems on Lotus ratio Lemma



§1 Problem Set Problem 1.1 (New Zealand MO 2020). Let ABCD be a square and let X be any point on side BC between B and C. Let Y be the point on line CD such that BX = Y D and D is between C and Y . Prove that the midpoint of XY lies on diagonal BD. Problem 1.2 (RMO 2012). Let ABC be a triangle and D be a point on the segment BC such that DC = 2BD. Let E be the mid-point of AC. Let AD and BE intersect in P . Determine the ratios BP : P E and AP : P D. Problem 1.3 (RMO 2012). Let ABC be a triangle. Let D, E be a points on the segment BC such that BD = DE = EC. Let F be the mid-point of AC. Let BF intersect AD in P and AE in Q respectively. Determine BP : P Q. Problem 1.4 (RMO 2012). Let ABC be a triangle. Let E be a point on the segment BC such that BE = 2EC. Let F be the mid-point of AC. Let BF intersect AE in Q. Determine BQ : QF . Problem 1.5 (2001 AMC 12 problem 22). In rectangle ABCD, points F and G lie on AB so that AF = F G = GB and E is the midpoint of DC. Also, AC intersects EF at H and EG at J. The area of the rectangle ABCD is 70. Find the area of triangle EHJ. E



D



J



H



A



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5 2



F



(B)



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35 12



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(C) 3



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7 2



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35 8



Problem 1.6 (RMO 1991 (Van Aubel’s Theorem)). Let P be an interior point of a triangle ABC and AP, BP, CP meet the sides BC, CA, AB in D, E, F respectively. Show that AF AE AP = + . PD F B EC Problem 1.7 (AIME I 2013 pr 3). Let ABCD be a square, and let E and F be points on AB and BC, respectively. The line through E parallel to BC and the line through F parallel to AB divide ABCD into two squares and two non square rectangles. The sum 9 AE of the areas of the two squares is 10 of the area of square ABCD. Find EB + EB AE Problem 1.8 (IOQM KVS 2021). Let D, E, F be points on the sides BC, CA, AB of a F E triangle ABC, respectively. Suppose AD, BE, CF are concurrent at P . If PP C = 23 , PP B = 2 PD m and = , where m, n are positive integers with gcd(m, n) = 1, find m + n 7 PA n Problem 1.9 (NMTC 2019 Junior’s P1). In a convex quadrilateral P QRS, the areas of triangles P QS, QRS and P QR are in the ratio 3 : 4 : 1. A line through Q cuts P R at A and RS at B such that P A : P R = RB : RS. Prove that A is the midpoint of P R and B is the midpoint of RS. Problem 1.10 (ISI TOMATO problem 451). In the figure that follows, BD = CD, of△ADH BE = DE, AP = P D and DG ∥ CF . Then compute area area of△ABC



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piyush kumar jha — March 18, 2023



Problems on Lotus ratio Lemma



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F P G H B



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