A triangle has consecutive side and height lengths for some integer n, as shown in image. What is its perimeter?
By solving the equation n^2 + (n+2-x)^2 = (n+3)^2 ---- 1 & n^2 + x^2 = (n+1)^2 ---- 2 We can get n = 12 and x = 5 as real solutions. Now if n = 12 and the base of the triangle = 14 is divided as 5 and 9 between the 2 right angled inner triangles. ie., The Pythagoras applied to first triangle is 12^2 + 5^2 = 13^2 = 144 + 25 = 169 and for the second triangle is 12^2 + 9^2 = 15^2 = 144 + 81 = 225
An acute triangle has a circumcircle, as shown in image. Minor arcs are reflected about each side of the triangle. Do the three reflected arcs intersect at the same point?
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A square is inscribed along the right angle of a right triangle, as shown, with vertical and horizontal lengths to the corners of 5 and 20. What is the area of the square?
Solve for the perimeter of a triangle whose altitudes have lengths of 170, 210, and 357.
