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Home » Circles » Inscribed Shapes » Circumscribed Circle

Circumscribed Circle

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

In geometry, "circumscribed" means "to draw around." A circumscribed circle is a circle that is drawn around a polygon so that it passes through all the vertices of a polygon inscribed in it.

All triangles have circumscribed circles, and in this lesson, we will devise a method to find that circle.

Problem

In triangle △ABC, D and E are the midpoints of sides AB and AC, respectively. DO and EO are perpendicular to AB and AC. Show that AO=BO=CO.

perp bisectors

Strategy

The simplest way to show line segments in a triangle are equal is with triangle congruency. Here, we have two pairs of triangles: (△AOE, △COE) and (△AOD, △BOD).

For each pair, we are given one equal side - since points E and D are the midpoints of their respective sides, and one equal angle - since DO and EO are perpendicular to each side, forming a linear pair of two right angles.

This makes it easy to show, that for each pair, the triangles are congruent, and by the transitive property of equality, AO=BO=CO.

Proof

(1) CE=EA //given, E is the midpoint of AC
(2) OE=OE //Reflexive property of equality, common side
(3) OE⊥AC //given
(4) m∠OEC=m∠OEA=90° //(3) , Linear pair formed by perpendicular lines
(5) △AOE≅ △COE //Side-Angle-Side postulate
(6) AO=CO //Corresponding sides of congruent triangles (CPCTC)
(7) BD=DA //given, D is the midpoint of AB
(8) OD=OD //Reflexive property of equality, common side
(9) OD⊥AB //given
(10) m∠ODB=m∠ODA=90° //(9), Linear pair formed by perpendicular lines
(11) △AOD≅ △BOD //Side-Angle-Side postulate
(12) AO=BO //Corresponding sides of congruent triangles (CPCTC)
(13) AO=BO=CO //(6),(12), Transitive property of equality

The point O is thus equidistant from all three of the triangle's vertices, so if we draw a circle by taking a compass, centering it on O and using OA as the radius, it will pass through A, B and C and we will have a circumscribed circle for triangle △ABC.

« Tangent-Secant Theorem
Distance Between the Centers of Overlapping Congruent Circles »

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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About

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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