jugoslav

[ΔABC] = 360 cm²

AD : DC = 1 : 2

BF : FC = 1 : 3

[ΔABF] = 1/4(360)

[ΔEBF] = 1/12(360) = 30 cm²

Triangle BCG is an isosceles right triangle.

BG and CG are the radii of a circle G.

(BC)² = 2r²            BC = r√2

[BXY] : [ABC] = 4 : 18

8(3x) = 14(2x - 2)

x = 7

No such triangle exists!!!  

YZ must be shorter than the sum of the lengths of XY and XZ!!!

A right-angled triangle has a perimeter of 1276 and the longest side of 533.  What is the length of the inradius?

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Side:       c = 533

(743 - a)² + a² = 533²

a = 308

b = 743 - 308 = 435

Semiperimeter:     s = 1276 / 2 = 638

Triangle area:        A = 1/2(308 * 435) = 66990

Inradius:            r = A / s = 105

[PAB] = 1/2(2 * 6)            

[QAB] = 1/2(2 * 4.5)

[PAB] : [QAB] = 4 : 3

BD = sqrt(8² + 6²)

DF = sin(29º) * 13

∠x = sin^-1(DF / BD)

Your answer is brilliant!!! Kudos!