MathLabs

Problem 1

Let ABCABC be an acute triangle inscribed in a circle Γ\Gamma. Let A1A_1 be the orthogonal projection of AA onto BCBC, so that AA1AA_1 is an altitude. Let B1B_1 and C1C_1 be the orthogonal projections of A1A_1 onto ABAB and ACAC, respectively. Point PP is such that quadrilateral AB1PC1AB_1PC_1 is convex and has the same area as triangle ABCABC. Is it possible that PP lies strictly in the interior of circle Γ\Gamma? Justify your answer.
Step 4 of 5: The equal-area locus of P
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{P:[AB1PC1]=[ABC]}=line through D parallel to B1C1\{P:[AB_1PC_1]=[ABC]\}=\text{line through }D\text{ parallel to }B_1C_1
Detailed analysis

Since B1,C1B_1,C_1 are fixed, the area of convex quadrilateral AB1PC1AB_1PC_1 is an affine function of the distance from PP to line B1C1B_1C_1. Hence the locus of points PP for which this area equals [ABC][ABC] is a line parallel to B1C1B_1C_1. Since DD already achieves this area by Step 3, that locus is exactly the line through DD parallel to B1C1B_1C_1, that is, the line through DD perpendicular to ADAD.