In ` A B C ,` as semicircle is inscribed, which lies on the side `cdot` If `x` is the lengthof the angle bisector through angle `C ,` then prove that the radius of the semicircle is `xsin(C/2)dot`

In a ΔABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius of the semicircle is (Where Δ is the area of the triangle ABC)

Let alt=blt=c be the lengths of the sides of a triangle. If a^2+b^2< c^2, then prove that angle C is obtuse .

In an acute angled triangle A B C , a semicircle with radius r_a is constructed with its base on BC and tangent to the other two sides. r_b a n dr_c are defined similarly. If r is the radius of the incircle of triangle ABC then prove that 2/r=1/(r_a)+1/(r_b)+1/(r_c)dot

Two chords A B and A C of a circle are equal. Prove that the centre of the circle lies on the angle bisector of /_B A Cdot

Two chords A B and A C of a circle are equal. Prove that the centre of the circle lies on the angle bisector of /_B A Cdot

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

If A ,\ B ,\ C are the interior angles of a triangle A B C , prove that tan(B+C)/2=cotA/2

If A ,B ,a n dC are three points on the hyperbola x y=c^2 such that A B subtends a right angle at C , then prove that A B is parallel to the normal to the hyperbola at point Cdot

If A ,B ,a n dC are three points on the hyperbola x y=c^2 such that A B subtends a right angle at C , then prove that A B is parallel to the normal to the hyperbola at point Cdot

If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot