MathDB
2023 Fall Theme p4B

Source:

December 23, 2023
2023FAlLthemegeo

Problem Statement

In triangle ABCABC, AB=13AB = 13, BC=14BC = 14, and CA=15CA = 15. Let MM be the midpoint of side ABAB, GG be the centroid of △ABC\triangle ABC, and EE be the foot of the altitude from AA to BCBC. Compute the area of quadrilateral GAMEGAME.
Proposed by Evin Liang
Solution. 23\boxed{23} Use coordinates with A=(0,12)A = (0,12), B=(5,0)B = (5,0), and C=(−9,0)C = (-9,0). Then M=(52,6)M = \left(\dfrac{5}{2},6\right) and E=(0,0)E = (0,0). By shoelace, the area of GAMEGAME is 23\boxed{23}.
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