47fc3c6761
* fix: clean-up Project Euler 281-300 * fix: missing image extension * fix: missing power Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing subscript Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
57 lines
1.8 KiB
Markdown
57 lines
1.8 KiB
Markdown
---
|
|
id: 5900f4971000cf542c50ffaa
|
|
title: 'Problem 299: Three similar triangles'
|
|
challengeType: 5
|
|
forumTopicId: 301951
|
|
dashedName: problem-299-three-similar-triangles
|
|
---
|
|
|
|
# --description--
|
|
|
|
Four points with integer coordinates are selected:
|
|
|
|
$A(a, 0)$, $B(b, 0)$, $C(0, c)$ and $D(0, d)$, with $0 < a < b$ and $0 < c < d$.
|
|
|
|
Point $P$, also with integer coordinates, is chosen on the line $AC$ so that the three triangles $ABP$, $CDP$ and $BDP$ are all similar.
|
|
|
|
<img class="img-responsive center-block" alt="points A, B, C, D and P creating three triangles: ABP, CDP, and BDP" src="https://cdn.freecodecamp.org/curriculum/project-euler/three-similar-triangles.gif" style="background-color: white; padding: 10px;">
|
|
|
|
It is easy to prove that the three triangles can be similar, only if $a = c$.
|
|
|
|
So, given that $a = c$, we are looking for triplets ($a$, $b$, $d$) such that at least one point $P$ (with integer coordinates) exists on $AC$, making the three triangles $ABP$, $CDP$ and $BDP$ all similar.
|
|
|
|
For example, if $(a, b, d) = (2, 3, 4)$, it can be easily verified that point $P(1, 1)$ satisfies the above condition. Note that the triplets (2,3,4) and (2,4,3) are considered as distinct, although point $P(1, 1)$ is common for both.
|
|
|
|
If $b + d < 100$, there are 92 distinct triplets ($a$, $b$, $d$) such that point $P$ exists.
|
|
|
|
If $b + d < 100\\,000$, there are 320471 distinct triplets ($a$, $b$, $d$) such that point $P$ exists.
|
|
|
|
If $b + d < 100\\,000\\,000$, how many distinct triplets ($a$, $b$, $d$) are there such that point $P$ exists?
|
|
|
|
# --hints--
|
|
|
|
`threeSimilarTriangles()` should return `549936643`.
|
|
|
|
```js
|
|
assert.strictEqual(threeSimilarTriangles(), 549936643);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function threeSimilarTriangles() {
|
|
|
|
return true;
|
|
}
|
|
|
|
threeSimilarTriangles();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|