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 2 of 5: The diameter through A is perpendicular to B1C1
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AD⊥B1C1AD\perp B_1C_1
Detailed analysis

Let OO be the circumcenter of ABCABC and let ADAD be the diameter of Γ\Gamma through AA. Since OA=OCOA=OC, triangle AOCAOC gives ∠DAC=90∘−∠B\angle DAC=90^\circ-\angle B. Step 1 gives B1C1∥BCB_1C_1\parallel BC, so AD⊥B1C1AD\perp B_1C_1.