In a right triangle ABC, a circle with a side. AB as diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.

In a right-triangle ABC in which angleB=90^(@) , a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC.

In the given figure, two circles touch each other externally at the point C. Prove that the common tangent to the circle at C bisects the common tangents at P and Q.

Prove that the tangents to a circle at the end points of a diameter are parallel.

If a, b, and c, are the sides of a right-angled triangle where C is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by r=(a+b-c)/(2) units.

ABC and ADC are two right triangle with common hypotenuse AC. Prove that CAD = CBD.

In a right angled triangle , square on the hypotenuse is equal to sum of the squares on the other sides. Prove the statement.

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

In the figure, an isosceles triangle ABC with AB=AC circumscribes a circle. Prove that the point of contact P bisects the base BC.

The equilateral triangles are drawn on the sides of a right triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of the triangles on the other two sides. OR In the given figure, PA, QB and RC are each perpendicular to AC. Prove that 1/x+1/z=1/y

In a right angled triangle, square on the hypotenuse is equal to sum of the squares on the other sides. Prove the statement.