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|dw:1435253588480:dw|

So, here I go:
First of I'll list up the hypothesis, which is the given information:
(H) ABC isoceles.
(T) AX = BX
|dw:1435253753194:dw|
To begin, I'll take the very definition of isoceles triangle:
\[(1) CA=CB\]
and :
\[(2)

the two base angles, say α, are equal so tanα = h/b1 = h/b2
where b1 + b2 = base therefore b1 = b2

show they are equal too for isoc

: )

ill allow the usage of AAS SSS and SAS only

the ancient greeks proved this 2000 years ago :)

yup

using tan -.- cheater face

thats a high end function

|dw:1435255353420:dw|

and so show the base angles of isosceles|dw:1435255424617:dw| are equal we can do

I really like your method Dan, Ill have that in my repertoire.

if u like these kind of examples i have a fun one for u

you can try to solve this problem with only geometry its a famous one dont look it up xD

Give me your best shot.

|dw:1435255751221:dw|

|dw:1435255859970:dw|

|dw:1435255900735:dw|

here u go :)

you must draw water from the river bank and go to the next city in the shortest route possible

is the question clear?

Yes

okay let a = 5, b= 10, L = 20 find point C|dw:1435256121850:dw|

|dw:1435260599276:dw|

|dw:1435260688944:dw|

isnt it pretty :)

Wow, much simpler than I thought.
And you used euclidean geometry... Reflections...

it's one of the most elegant solutions from history

With no wonder.
Very straight and elegantly solved.

especially nowadays when everyone just resorts to calculus

this one can be solved as a minimization problem, and gets all messy

and an interesting thing comes which is that these angles are equal

|dw:1435261128090:dw|

so the point always such that those angles are equal, so these 2 triangles are similiar

and the main triangle here can be draw too to solve completely

|dw:1435261208966:dw|

|dw:1435261234039:dw|

proportionality.
I really need to dig back to geometry, I am starting to get rusty.

theres multiple ways to solve it but this is the mains solution |dw:1435261434618:dw|

once u draw this pic u are pretty much done technically, as u just managed to draw shortest line

in books, they usually stop there

there an extension to this, which is in higher dimension using same idea

or like if u have to hit 2 lines

In 3 dimensions?

|dw:1435261603447:dw|

like this one, very similar , but now u have to hit 2 lines before going from town a to b

maybe u can find this one :) and get back to me tomorrow

I will try, I believe you have many questions to answer nowdays :x