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Problem 509

Show that in a quadrilateral circumscribed about a circle, the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides.

Expert verified

In a quadrilateral ABCD circumscribed about a circle, let the circle touch the sides AB, BC, CD, and DA at points P, Q, R, and S, respectively. Using tangent properties and the tangent-tangent theorem, we find equalities for the lengths of the segments: AP = AS, BQ = BP, CR = CQ, and DS = DR. Now, summing up the opposite sides, we get (AB + CD) = (x + y + u + v) and (AD + BC) = (s + t + z + w), where x, y, z, w, u, v, s, and t represent the lengths of the segments. Substituting the equal lengths found, we arrive at the conclusion that the sum of the lengths of a pair of opposite sides equals the sum of the lengths of the remaining pair of opposite sides in a quadrilateral circumscribed about a circle.

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Chapter 26

Given: \(\triangle \mathrm{ABC}\) with sides of length \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\). The inscribed circle, \(Q\), intersects \(\underline{A B}\) at \(D, \underline{B C}\) at \(E\), and \(\underline{C A}\) at \(F\). Find the lengths \(\mathrm{AD}\) and \(\mathrm{DB}\) in terms of \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\).

Chapter 26

Find the radius of a circle inscribed in a triangle whose sides have lengths 3,4 and 5 .

Chapter 26

Find the radius, \(\mathrm{r}\), of the inscribed circle of right triangle \(\mathrm{ABC}\) in terms of leg lengths a and \(\mathrm{b}\) and hypotenuse length \(\mathrm{c}\).

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