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Home » Circles » Inscribed Shapes » Circle Inscribed in a Quadrilateral

Circle Inscribed in a Quadrilateral

Last updated: Sep 30, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A tangential quadrilateral is a quadrilateral whose four sides are all tangent to a circle inscribed within it. In such a quadrilateral, the sum of lengths of the two opposite sides of the quadrilateral is equal.

This is known as the Pitot theorem, named after Henri Pitot, a French engineer who proved it in the 18th century.

Problem

A circle is inscribed in a quadrilateral ABCD. Show that |AB|+|CD| = |BC|+|DA|

circle inscribed in a quadrilateral

Strategy

We are working with a tangential quadrilateral, so all the sides are tangent to the circle. Let's review the properties of tangents to a circle.

One of the theorems we have proved is the two tangent theorem - two tangents from the same point outside a circle have equal lengths to the points of tangency. This should be useful here, as it deals with the lengths of such tangents.

Looking at the quadrilateral, we have four such points outside the circle. Each one of the quadrilateral's vertices is a point from which we drew two tangents to the circle.

Applying the two tangent theorem to each one of these points, we get 4 pairs of equal-size line segments: AP=AS=a; BP=BQ=b; CQ=CR=c and DR=DS=d.

four cases of The Two Tangent Theorem

And in each of the opposite pairs of sides of the quadrilateral, we have exactly one of these segments, so their sums are equal. And we have proven the Pitot theorem for a circle inscribed in a quadrilateral.

Proof

(1) AP=AS=a // Two tangent theorem
(2) BP=BQ=b // Two tangent theorem
(3) CQ=CR=c // Two tangent theorem
(4) DR=DS=d // Two tangent theorem
(5) |AB|+|CD| = |AP|+|PB|+|CR|+|RD|= a+b+c+d
(6) |BC|+|DA| = |BQ|+|QC|+|DS|+|SA|=b+c+d+a=a+b+c+d // Commutative property of addition
(7) |AB|+|CD| = |BC|+|DA| //(5),(6), transitive property of equality

« Consecutive Interior Angles Converse Theorem
Area of Parallelogram Given Diagonals and a Side »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected].

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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