Originally Posted by **ticbol**

Zeez, I think your computer's monitor is upside down. In your drawing as posted, the y's are increasing downwards. Umm, cannot be upside down because the x's are increasing rightwards which is normal.

How did you manage to do that? Amazing.

Normal?

Your normals here are lines normal/perpendicular to the circumferences of the two equal circles. So being normal to the circumferences, they pass through the centers of the circles, correspondingly.

So you found the coordinates of the point of normalcy of line N2-N2' to be (400,330).

Actually, it is (400.794,330).

(x-522)^2 +(330-377)^2 = (130)^2

(x-522)^2 = 16,900 -(-47)^2 = 14,691

x-522 = +,-sqrt(14,691) = +,-121.206

x = -121.206 +522 = 400.794 -----------***

Then, the attack could be:

---get the equation of the line passing through the sought-for-(x,y), (400.794,330), and (600,355) points,

---get the intersection of that said line with the circle whose center is at (338,377) to get the (x,y).

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The line passing through (400.794,330), (x,y), (600,355):

slope, m = (355 -330)/(600 -400.794) = 0.1255

Using the point (600,355), the point-slope form of the equation of the said line is

y-355 = (0.1255)(x-600)

y = 0.1255x -75.3 +355

y = 0.1255x +279.7 ----------(1)

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The equation of the circle whose center is at (338,377) and whose radius is 130:

(x-338)^2 +(y-377)^2 = (130)^2 -------(2)

The intersection of line (1) and circle (2) is point (x,y).

At that intersection, the coordinates of (1) and (2) are the same, so substitute the y of (1) into (2),

(x-338)^2 +(0.1255x +279.7 -377)^2 = (130)^2

(x-338)^2 +(0.1255x -97.3)^2 = 16,900

Solved the problem. Just simplify, etc.

Here goes,

[x^2 -676x +114,244] +[0.01575x^2 -24.422x +9467.29] -16,900 = 0

1.01575x^2 -700.422x +106,811.29 = 0

Divide both sides by 1.01575,

x^2 -689.561x +105,155.097 = 0

Use the Quadratic Formula,

x = {-(-689.561) +,-sqrt[(-689.561)^2 -4(1)(105,155.097)]} /(2*1)

x = {689.561 +,-234.252} /2

x = 461.906 or 227.654

On the drawing, x is to the right of the center of the circle, so,

x = 461.906 -----------***

And, substitute that into (1),

y = 0.1255(461.906) +279.7 = 337.67 ----------***

Therefore, (x,y) is (461.906,337.67) --------------------answer.