Inscribed Angle Roman Kvasov
Basic Definitions and Properties 1) Inscribed angle is the angle with the vertex and the endpoints on the circle’s circumference 2) All inscribed angles with the vertices lying in the same semi-plane and endpoints being the endpoints of the equal length chords are equal 3) Central angle is the angle with the vertex in the centre of the circle and the endpoints on the circle’s circumference 4) The central angle is twice larger than the inscribed angle with the same endpoints and vertices lying in the same semi-plane 5) The inscribed angle with the endpoints being the endpoints of some diameter is right
Practice Problems 1.
Quadrilateral ABCD is inscribed into a circle. K is the point of the intersection of its diagonals. The circle passing through points A , B and K intersects BC and AD at M and N . Prove KM = KN .
2.
In the circle with radius R , AB and CD are two perpendicular chords. Prove that AC + BD = 4 R .
3.
Rectangle ABCD is inscribed into a circle. Prove that for any point X of this circle it is true that
2
2
2
AX 2 + BX 2 + CX 2 + DX 2 is a constant. 4.
The altitudes of the acute triangle ABC intersect with its circumcircle at points H1 , H 2 at H 3 . Prove that the altitudes of the triangle ABC are the angle bisectors of the triangle H1 H 2 H 3 .
5.
The angle bisector of the exterior angle C of the triangle ABC intersects its circumcircle at point D . Prove that AD = BD .
6.
On the chord AB of the circle ω with center O one chose the point C . The circumcircle of the triangle AOC intersects the circle ω at the point D . Prove that BC = CD .
7.
Two circles intersect at the points A and B . Through some point K on the first circle one draws two lines KA and KB that intersect the second circle at points P and Q respectively. Prove that PQ is perpendicular to the diameter of the first circle passing through point K .
8.
Circles ω1 and ω2 intersect at A and B . The center O of ω1 lies on ω2 . Chord AC of ω1 intersects
ω2 at point D such that B and O lie in different semi-planes with respect to AC . Show OD ⊥ BC . 9.
D
Angle A of the triangle ABC is 60 . The angle bisector of the angle A intersects the circumcircle at point M . On the side AB one chose point K such that KM = BM . Prove that AM bisects KC .
10. AD is the angle bisector of the triangle ABC . The perpendicular bisector to AD intersects BC at the point N . Prove that NA is tangent to the circumcircle of the triangle ABC .