☰ Contents · Geometry

The inscribed and circumscribed circle

Lessons 38–39 · 2 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
38

The inscribed circle

Textbook: pp. 118–120
GoalKnow the properties of an incircle and of a circumscribed polygon.
New words
incircle · ichki chizilgan aylanacircumscribed polygon · tashqi chizilgan ko‘pburchakangle bisector · bissektrisaopposite sides · qarama-qarshi tomonlar
Explanation

If every side of a polygon touches a circle, the polygon is circumscribed about the circle and the circle is inscribed in the polygon. The centre of an inscribed circle is equally far from all sides. Every triangle has exactly one incircle; its centre is the intersection point of the angle bisectors, since a point equally far from the sides of an angle lies on its bisector. In a circumscribed quadrilateral the sums of opposite sides are equal: AB + CD = AD + BC. The proof uses the equality of the tangent segments from one point: add AE = AQ, BE = BF, CP = CF, DP = DQ. The converse holds too: a convex quadrilateral whose opposite sides have equal sums has an inscribed circle.

Worked examples
The distances from the vertices to the points of tangency are 3, 4, 7. The sides are 3 + 4 = 7, 4 + 7 = 11, 3 + 7 = 10; the perimeter is 28.
In a circumscribed ABCD with AB = 9, BC = 7, CD = 12: AD = 9 + 12 − 7 = 14.
Class activity

“Where bisectors meet”: draw a triangle, draw two angle bisectors with a protractor and measure the distances from their intersection to the sides to check they are equal.

Practice
1
The distances from the vertices of a triangle to the points of tangency are 2, 5 and 6. What is the perimeter?
2
In a circumscribed quadrilateral AB = 8, CD = 11, BC = 9. How long is AD?
3
A trapezoid circumscribed about a circle has legs 6 and 10. How long is its midline?
4
Why does the centre of the incircle lie on the angle bisectors of the triangle?