Problems

Prove that from the point $C$ lying outside of the circle we can draw exactly two tangents to the circle and the lengths of these tangents (that is, the distance from $C$ to the points of tangency) are equal.

Two circles intersect at points $A$ and $B$. Point $X$ lies on the line $AB$, but not on the segment $AB$. Prove that the lengths of all of the tangents drawn from $X$ to the circles are equal.

Let $a$ and $b$ be the lengths of the sides of a right-angled triangle and $c$ the length of its hypotenuse. Prove that:

a) The radius of the inscribed circle of the triangle is $(a+b-c)/2$;

b) The radius of the circle that is tangent to the hypotenuse and the extensions of the sides of the triangle, is equal to $(a+b+c)/2$.

Let $G,F,H$ and $I$ be the midpoints of the sides $CD,DA,AB,BC$ of the square $ABCD$, whose area is equal to $S$. Find the area of the quadrilateral formed by the straight lines $BG,DH,AF,CE$.

a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.

b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.

Prove that the bisectors of a triangle intersect at one point.

Prove that the following inequalities hold for the Brockard angle $\phi$:

a) ${\phi }^3\le (\alpha -\phi )(\beta -\phi )(\gamma -\phi )$ ;

b) $8{\phi }^3\le \alpha \beta \gamma$ (the Jiff inequality).

Prove that a convex quadrilateral $ABCD$ can be drawn inside a circle if and only if $\mathrm{\angle }ABC+\mathrm{\angle }CDA={180}^{\circ }$.

a) Find the locus of the points that are equidistant from two parallel lines.

b) Find the locus of the points that are equidistant from two intersecting lines.

Find the locus of the midpoints of the segments, the ends of which are found on two given parallel lines.