Two acute angled traingle ABC and PQR are similar. Their circumcentres are X,Y respectively. If BC and QR are corresponding sides, then prove that BX: QY = BC:QR.

DeltaABC and DeltaPQR are two similar acute triangles. Their circumcentres are X and Y respectively. If BC and QR be two corresponding sides, then prove that BX:QY=BC:QR .

If in ABC,A is right angle and BP and CQ are two medians, then prove that 5 BC^2 = 4 (BP^2 + CQ^2) .

O is the circumcentre of Delta ABC and OD is perpendicular on the side BC, prove that angle BOD= angle BAC

ABC is a right-angled triangle of which /_A = right angle. P and Q are two points on AB and AC respectively.By joining P,Q,B,Q and C,P prove that BQ^(2) + PC^(2) = BC^(2) + PQ^(2)

Two circles drawn with centres A and B touch each other externally at C, O is a point on the tangent drawn at C, OD and OE are tangents drawn to the circles of centre A and B respectively. If angle COD=56^(@), angle COE=40^(@), angle ACD=x^(0) and angle BCE =y^(@), then prove that OC = OD =OE and x-y=8.

If the lengths of the sides vec(BC), vec(CA) and vec(AB) of the triangle ABC be a:b:c respectively and the lengths of perpendicular from the circumcenter upon vec(BC), vec(CA) and vec(AB) be x, y, z respectively, then prove that, a/x + b/y + c/z = (abc)/(4xyz) .

In the adjacent figure 'P' is any arbitrary interior point of the triangle ABC. H_a,H_b and H_c are the length of altitudes draw from vertices A,B and C respectively. If x_a,x_b and x_c represent the distance of 'P' from sides BC, CA and AB respectively, then the minimum value of (H_a/x_a+H_b/x_b+H_c/x_c) must be

(i) If the cosines of two of the angles of a triangle, which is not isosceles, are iversely proportional to the corresponding opposite sides, show that the triangle is right angled. (ii) In a triangle ABC, If A=2B then show that, a^(2)- b^(2)=bc .

In an isosceles right-angled triangle ABC, /_B=90^(@) . The bisector of /_BAC intersects the side BC at the point D. Prove that CD^(2) =2BD^(2)