Tangency of Circles | Problems 16-21 | Problem Solving | Volchkevich | Geometry Lessons 7-8 Grades
16. A circle is inscribed in a triangle. One of the points of tangency
is connected to the opposite vertex. The resulting segment
divides the triangle into two triangles. Prove
that their inscribed circles are tangent.
17. Two sides of a triangle are 2 and 3. A line is drawn through their common
vertex such that the circles marked in the figure are tangent. In what ratio does this line divide the third
side of the triangle, if this side is 4?
18. Three circles are externally tangent to each other. Prove that the circle passing through the three points of tangency is inscribed in the triangle formed by the centers of the circles.
19. Construct three circles that are externally tangent to each other at three given points.
20. Construct three circles, two of which are externally tangent at point A, and internally tangent to the third circle at points B and C.
*21. Two circles touch externally at point B, and
at points A and C they touch a third circle internally. It turns out
that the radius of the third circle is equal to the radius of the circle passing through points A, B, and C. Find angle ABC.