In triangle PQR, PQ=QR and RS is the altitude. PR is extended to point T such that QT=10.

The values of tan ∠RQT, tan ∠SQT and tan ∠PQT form a geometric sequence, and:
The values of cot ∠SQT, cot ∠RQT and cot ∠SQR form an arithmetic sequence.

Determine the area of the triangle PQR.

With altitude QS…

Let a = tan/RQT,    b = tan/SQR = tan/PQS,    c = tan/SQT .

Since /SQT = /RQT + /SQR,        c = (a + b)/(1 – ab)        (1)

Since /PQT = /PQS  + /SQT,        tan/PQT = (b + c)/(1 - bc)

The geometric sequence:  a,  c,  (b + c)/(1 - bc)  gives:

                        a(b + c)/(1 – bc) = c²                             (2)

and the arithmetic sequence:  1/c,  1/a,  1/b  gives:

                        1/b + 1/c = 2/a                                      (3)

Using (2) and (3),   a(b + c) = c²(1 – bc) = 2bc

which gives                    b = c/(2 + c²)                            (4)

and using (3)                 a = 2c/(3 + c²)                           (5)

Substituting (4) and (5) in (1) then gives (eventually) c = 1,
from which  a = ½ and b = 1/3.

So triangle SQT is isosceles with QS = 5*sqrt(2).

Area PQR = (QS)²*tan/SQR = 50*1/3  =  50/3

Posted by Harry on 2014-08-20 21:32:31