Pretty basic Pythagorean & Trigonometry

Nov 2019
1
0
france
Hi,

It's probably gonna be a pretty simple thing to solve for you but I'm not that good in math so I'm really thankful for any help that you can bring me.

Long story short (refer picture below):

As you can see we have two circles forming two right angled triangles with a third one. The information in white are the ones I know and the ones in the red are the ones I'm looking for. So given the coordinates a and b, and the lengths ab, bc and ac, is it possible to find the coordinates of the point c in the right angled triangle abc.
Then, if it's possible to have the coordinate of c, is it possible to find the coordinate of the point d with the information we have?

HD-2472430_Screen Shot 2019-11-10 at 11.27.36 PM.png):
 
Dec 2014
134
103
USA
Yes, it is possible. Start by using Pythagoras to determine the length of $ac$.

Note the radius of the red circle is $ab-bc$, call it $r$.

Determine the angle of the yellow triangle at point $b$ using an inverse trig function from triangle $abc$.

Let the radius of the circle centered at point $d$ be $R$.

Use the cosine law to set up an equation to solve for $R$:
$(R+r)^2 = (ab)^2+(bc+R)^2 -2(ab)(bc+R) \cdot \dfrac{bc}{ab}$

At this point, you have the lengths of all three sides of triangle $abd$.

From here, you should be able to determine points $c$ and $d$ using a few principles of analytic geometry which I am to lazy to do at this time. At least you have a start.
 
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Jun 2013
1,113
590
Lebanon
Hi,

It's probably gonna be a pretty simple thing to solve for you but I'm not that good in math so I'm really thankful for any help that you can bring me.

Long story short (refer picture below):

As you can see we have two circles forming two right angled triangles with a third one. The information in white are the ones I know and the ones in the red are the ones I'm looking for. So given the coordinates a and b, and the lengths ab, bc and ac, is it possible to find the coordinates of the point c in the right angled triangle abc.
Then, if it's possible to have the coordinate of c, is it possible to find the coordinate of the point d with the information we have?

View attachment 39585):
in the first question,
given a right triangle \(\displaystyle abc\),
given the coordinates of the points \(\displaystyle a,b\), and
given the lengths of the sides of triangle \(\displaystyle abc\)
Find the coordinates of the point \(\displaystyle c\)
(No need for any circles to do this part of the problem)

For the second part,
I understand that we have to find the coordinates of the point \(\displaystyle d\)
but I don't understand the given