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in Olympiad by (71.4k points)

Let ABCD be a convex quadrilateral. Let the diagonals AC and BD intersect in P. Let PE, PF, PG and PH be the altitudes from P on to the sides AB, BC, CD and DA respectively. Show that ABCD has an in circle if and only if

1/PE + 1/PG = 1/PF + 1/PH

1 Answer

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Best answer

Let AC and YZ intersect at Q then ∠QYC = ∠QWD = Q(say)

(As ∠ made by tangent at a point is same as ∠ made by chord through point in alternate segment)

=> AC, XZ and YW are concurrent similarly BD, XZ and YW are concurrent

Hence AC, BD, YW and XZ are concurrent

=> Q = P

Let Area of MNJ = [MNJ]

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