PQR is an acute angled triangle.
It is known that:
- S, T and U are points located respectively on the sides QR, RP and PQ.
- PS is perpendicular to QR.
- RU is the internal bisector of the angle PRQ.
- RU intersects PS and ST respectively at the points J and K.
- PT = TR.
- UJ=2, JK = 1 and KR = 3
Determine the perimeter of the triangle PQR.
Since K bisects UR and T bisects PR, TK is parallel to PQ. It follows that S bisects QR. So PS bisects angle QPR and the triangle is at least isosceles with apex at P and base QR.
Playing around with Geometer's Sketchpad, the triangle seems it must be uniquely equilateral. That is, only one position for Q relative to PR works, and since an equilateral triangle does work analytically, that is the unique solution.
Half of one side of the triangle comes out to 2*sqrt(3), so the full perimeter is 12*sqrt(3) ~= 20.78460969082653.
Posted by Charlie
on 2007-09-28 11:05:16