{"id":331,"date":"2020-02-18T12:02:44","date_gmt":"2020-02-18T12:02:44","guid":{"rendered":"https:\/\/potatodie.nl\/diffuse-write-ups\/?p=331"},"modified":"2020-02-24T14:34:44","modified_gmt":"2020-02-24T14:34:44","slug":"angle-bisector-theorem","status":"publish","type":"post","link":"https:\/\/potatodie.nl\/diffuse-write-ups\/angle-bisector-theorem\/","title":{"rendered":"Angle bisector theorem"},"content":{"rendered":"\n

Imagine a triangle.<\/p>\n\n\n\n

\"Triangle<\/figure>\n\n\n\n

Let’s divide the angle at point A<\/em> in half.<\/p>\n\n\n\n

\"Bisecting
Bisecting the angle at point A<\/em> of \u25b3ABC<\/em> .<\/figcaption><\/figure>\n\n\n\n

The dotted line is called the bisector<\/em>. The bisector divides the opposite line segment in a part of length x<\/em> and a part of length y.<\/em><\/p>\n\n\n\n

Equal ratios<\/h4>\n\n\n\n

The Angle Bisector Theorem states that the ratio x : b<\/em> equals the ratio y : c<\/em>. Or equivalently, b\/x = c\/y.<\/em> This means that if b <\/em>is for example twice the length of x<\/em>, then c<\/em> is twice the length of y.<\/em> <\/p>\n\n\n\n

In the drawing below, measuring in millimeters, I found b\/x<\/em> = 32\/25 (= 1.28), while c\/y<\/em> = 61\/48, which is roughly1.27. <\/em>So my drawing is not so bad in that respect.<\/p>\n\n\n\n

\"Example
Illustrating the theorem with a measurement check.<\/figcaption><\/figure>\n\n\n\n

Who needs proof?<\/h4>\n\n\n\n

Several proofs of this theorem exist. My interest however is to find a construction that makes immediately clear that the theorem holds. I just want to see<\/em> that it is true. And from there, effortlessly I hope, write down the proof if anyone needs it.<\/p>\n\n\n\n

First attempt: Pairs of congruent triangles<\/h4>\n\n\n\n

Use compass and ruler (if you like) to construct points P, Q<\/em> and R<\/em>.<\/p>\n\n\n\n

\"Constructing
Constructing triangles, with several congruencies.<\/figcaption><\/figure>\n\n\n\n

There are two pairs of congruent triangles: \u25b3ABR<\/em> (the large triangle containing all geometry) to \u25b3APC<\/em> (the little one crouching at the top) and \u25b3PCQ<\/em> to \u25b3RBQ<\/em> (the green ones). From these it follows that x : y = PC : BR = b : c<\/em>. <\/p>\n\n\n\n

So x : b = y : c.<\/em><\/p>\n\n\n\n

It’s not complicated, but it requires too many steps. I would like something simpler.<\/p>\n\n\n\n

Second attempt: One congruency less<\/h4>\n\n\n\n
\"Construction
Bisecting \u25b3ABC<\/em> and expanding one part so it becomes congruent with the other.<\/figcaption><\/figure>\n\n\n\n

Another possibility is to extend \u25b3ABD<\/em> with the isosceles triangle \u25b3BDE<\/em> as above. The two coloured triangles have two angles in common (indicated with open and closed discs) and so are congruent.<\/p>\n\n\n\n

So now we have congruent triangles with x<\/em> and b<\/em> in one triangle and y<\/em> and c<\/em> in the other. Now it’s obvious that x : b = y : c.<\/em> <\/p>\n\n\n\n

Third attempt: Simpler congruency<\/h4>\n\n\n\n

Here’s another, even simpler one, arriving at two triangles with two equal angles, therefor congruent.<\/p>\n\n\n\n

\"Congruence
Bisecting \u25b3ABC<\/em> and expanding one part so it becomes an isosceles triangle. <\/figcaption><\/figure>\n\n\n\n

We got rid of the reflection. The congruency consists of a scaling and a rotation operation only.<\/p>\n\n\n\n

Final attempt<\/h4>\n\n\n\n

In the previous attempt it appeared the reflection part of the congruency wasn’t needed. Well, the rotation isn’t needed either:<\/p>\n\n\n\n

\"Simple
Congruent triangles lined up.<\/figcaption><\/figure>\n\n\n\n

No rotation or reflection, just a scaled (and perhaps translated \u2013 depends on the transformation origin) copy.<\/p>\n\n\n\n

And the winner is…<\/h4>\n\n\n\n

I’m satisfied with that, but if you find alternatives, please let me know! Actually, I think my preference goes to attempt number 3.<\/p>\n","protected":false},"excerpt":{"rendered":"

Intuitive approach to proof of angle bisector theorem<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30,29],"tags":[],"class_list":["post-331","post","type-post","status-publish","format-standard","hentry","category-geometry","category-mathematics"],"acf":[],"_links":{"self":[{"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/posts\/331","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/comments?post=331"}],"version-history":[{"count":17,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/posts\/331\/revisions"}],"predecessor-version":[{"id":382,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/posts\/331\/revisions\/382"}],"wp:attachment":[{"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/media?parent=331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/categories?post=331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/potatodie.nl\/diffuse-write-ups\/wp-json\/wp\/v2\/tags?post=331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}