Math Help - Prove Points are Concyclic.

1. Prove Points are Concyclic.

Hey Everyone,

I have a question here that asks:

Through a point on the diagonal of a square, lines PR, QS are drawn parallel to the sides, P, Q, R, S being on the sides. Prove that these four points are concyclic.

Any help would be greatly appreciated. Thanks everyone.

2. Let O be the point of intersection of the lines PR and QS (which is on the diagonal of the square). Then let the measure of OPS = a and add the measures of the following angles:

OPS + OPQ = SPQ = a + 45

OQP + OQR = PQR = 45 + a

ORQ + ORS = QRS = (90 - a) + 45 = 135 - a

OSR + OSP = RSP = 45 + (90 - a) = 135 - a

Then SPQ + QRS = 180 and PQR + RSP = 180, proving that PQRS is a cyclic quadrilateral (i.e., P, Q, R, S are concyclic).

3. Hello, GreenDay14!

I have an analytic proof . . .

Through a point on the diagonal of a square,
lines $PR, QS$ are drawn parallel to the sides; $P, Q, R, S$ being on the sides.
Prove that these four points are concyclic.
Code:
|             Q
(0,a)|           (b,a)    (a,a)
͸ - - - - - - o - - ͸
|             |   * |
P   |             | *   |   R
(0,b) o - - - - - - * - - o (a,b)
|           * |     |
|         o   |     |
|       * C   |     |
|     *       |     |
|   *         |     |
| *           |     |
- - - ͸ - - - - - - o - - ͸ - - -
(0,0)           (b,0)    (a,0)
S

Let $a$ = side of the square.

Let $(b,b)$ be the point on the diagonal.

Then we have: . $P(0,b),\;Q(b,a),\;R(a,b),\;S(b,0)$

Let $C\left(\tfrac{a}{2},\:\tfrac{a}{2}\right)$ be the center of the square.

Find the distances: . $CP,\:CQ,\:CR,\:CS$

. . $CP^2 \:=\:\left(\tfrac{a}{2}-0\right)^2 + \left(\tfrac{a}{2}-b\right)^2$

. . $CQ^2 \:=\:\left(\tfrac{a}{2}-b\right)^2 + \left(a - \tfrac{a}{2}\right)^2$

. . $CR^2 \:=\:\left(\tfrac{a}{2}-a\right)^2 + \left(\tfrac{a}{2}-b\right)^2$

. . $CS^2 \:=\:\left(\tfrac{a}{2}-b)\right)^2 + \left(\tfrac{a}{2} - 0\right)^2$

We find that they are all equal to: . $\tfrac{1}{2}a^2 - ab + b^2$

Hence, there is a point $C$ equidistant from $P,Q,R,S.$
. . $C$ is the circumcenter of quadrilateral $PQRS.$

Therefore, $P,Q,R,S$ are concyclic.