A convex quadrilateral ABCD is inscribed and circumscribed. If the

Peia6tvsr

Peia6tvsr

Answered question

2022-05-10

A convex quadrilateral ABCD is inscribed and circumscribed. If the diagonals AC and BD are perpendicular, show that one of them divides the quadrilateral into two congruent right triangles.

Answer & Explanation

Makhi Lyons

Makhi Lyons

Beginner2022-05-11Added 15 answers

Step 1
Let the diagonals intersect at O and set: a = A O , c = C O , d = D O .
The sums of opposite sides must be the same, giving:
a 2 + b 2 + c 2 + d 2 = a 2 + d 2 + b 2 + c 2 .
Squaring and simplifying we get:
( a 2 + b 2 ) ( c 2 + d 2 ) = ( a 2 + d 2 ) ( b 2 + c 2 ) , ,
which reduces to
( a 2 c 2 ) ( b 2 d 2 ) = 0. .
Hence, a = c or b = d , that is one of the diagonals bisects the other one and is thus a diameter of the circumscribed circle.

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