Question

let two circles intersect each other at point A and D.Let the diameter
AB intersect the circle with centre P at point N and diameter AC
intersect the circle in point M with
centre Q. Then prove that
AC × AM = AB × AN​

Answer 1

AC × AM = AB × AN​

Step-by-step explanation:

In Δ ACN and Δ ABM we have

∠MAB=∠NAC(Vertically opposite angle)

∠AMB=∠ANC( semi circle angle is a right angle)

∴ Δ ACN ≈ Δ ABM (By AA Similarity)

i.e. AC/AB=AN/AM

Hence ,proved  that

AC × AM = AB × AN​

Answer 2

Given :

Two circles intersect each other at point A and D

The diameter AB intersect the circle with centre P at Point N

And diameter Ac intersect the circle at point B along with centre Q

To find :

Proof = C × AM = AB × AN​  ?

Solution :

Let us assume that

AB and AC are circle diameters

As We know that,

Angle inscribed in semi-circle is 90°

Therefore  

∠AMB =∠ANC = 90°

 In △ACN and △ABM,  there are following conditions

∠MAB = ∠NAC  represent Vertically opposite angle

∠AMB = ∠ANC  represent angle in the semi-circle is to be right angle

Therefore  △ACN ∼△ABM  represent Angle to Angle similarity test

Now

$\frac{AC}{AB} = \frac{AN}{AM}$

hence,

The $AC \times AM = AB \times AN$ is proved

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