Question

Prove Ptolemy's Theorem through Algebra only

Answer 1

@Directrix @jim_thompson5910

Answer 2

take a look here
http://www.cut-the-knot.org/Curriculum/Geometry/HingedPtolemy.shtml

Answer 3

it doesn't work and it isn't exactly all algebra

Answer 4

There is a nice one here with sines and cosines

http://www.cut-the-knot.org/proofs/sine_cosine.shtml

Answer 5

that whole site is pretty good, take a look around

Answer 6

im not allowed to use trig sorry

Answer 7

This one is good too, By Inversion

http://www.cut-the-knot.org/proofs/PtolemyByInversion.shtml

Answer 8

@DanJS I said I cant use trig and anything over year 8

Answer 9

here is a simple one, scroll down to where it says proof, it is just by similar triangles

http://www.cut-the-knot.org/proofs/ptolemy.shtml

Answer 10

@DanJS I said im ONLY allowed to use ALGEBRA

Answer 11

On the diagonal BD locate a point M such that angles ACB and MCD be equal. Since angles BAC and BDC subtend the same arc, they are equal. Therefore, triangles ABC and DMC are similar. Thus we get CD/MD = AC/AB, or AB·CD = AC·MD.

Now, angles BCM and ACD are also equal; so triangles BCM and ACD are similar which leads to BC/BM = AC/AD, or BC·AD = AC·BM. Summing up the two identities we obtain

AB·CD + BC·AD = AC·MD + AC·BM = AC·BD

Answer 12

reload

Answer 13

that is just algebra, not sure if you can use that one

Answer 14

asi in im not supposed to use angles and similarity etc.

Answer 15

@TuringTest can u help

Answer 16

hmm, sorry i really dont know any simpler one... here is one other i found that may help, it uses angles though
http://www.trans4mind.com/personal_development/mathematics/geometry/ptolemy_theorem.htm

Answer 17

yeah, i cant think of a proof that doesnt involve similar triangles. ...
http://planetmath.org/ProofOfPtolemysTheorem
Same as last one, just simpler

Answer 18

good luck