# Cyclic Quadrilateral

If ABCD is a cyclic quadrilateral, then the sum of opposite angles is 180 degrees. It means, ∠A + ∠C = ∠B + ∠D = 180 degrees.

## Product of Diagonals : Ptolemy Theorem

In a cyclic quadrilateral, the sum of product of two pairs of opposite sides equals the product of two diagonals. This property of cyclic quadrilateral is known as **PTOLEMY THEOREM**.

If ABCD is a cyclic quadrilateral, then **AB x CD + AB x BC = AC x BD**.

**Proof:**

Take a point M on BD so that ∠ACB = ∠MCD.

As we know that the angles in same segment are equal. Then, ∠BAC = ∠BDC.

Now by AA Similarity, we have ∆ACB ~ ∆DCM.

Thus, we get ⇒ AB x CD = AC x DM … (1)

Now, ∠DAC = ∠DBC and ∠DCA = ∠BCM, then by AA similarity, we have ∆ACD ~ ∆BCM.

Thus, we get ⇒ AD x BC = AC x BM … (2)

From (1) + (2), we get **AB x CD + AD x BC = AC x BD**.

## Ratio of Diagonals

In a cyclic quadrilateral, the ratio of the diagonals equals the ratio of the sum of products of sides that share the diagonal’s end points.

If ABCD is a cyclic quadrilateral, then .

**Proof:**

Let ABCD is a cyclic quadrilateral whose diagonals intersect at P.

As we know that the angles in same segment are equal. Then, ∠BAC = ∠BDC and ∠ADB = ∠ACB.

From vertically opposite angles, ∠APB = ∠CPD and ∠APD = ∠BPC.

Now, by AA similarity, we have ∆PAD ~ ∆PBC.

Then, we get .

From first two ratios, ⇒

Multiplying both sides by AB, we get

⇒ … (1)

From last two ratios, ⇒

Multiplying both sides by CD, we get

⇒ … (2)

Now again, by AA similarity, we have ∆PAB ~ ∆PDC.

Then, we get .

From first two ratios, ⇒

Multiplying both sides by AD, we get

⇒ … (3)

From (1), (2), (3), we get .

Since ,

Adding (1st and 3rd Ratios) and (2nd and 4th Ratios)

⇒

Thus, we get