51 lines
1.4 KiB
Markdown
51 lines
1.4 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.
|
|
|
|
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--
|
|
|
|
`euler299()` should return 549936643.
|
|
|
|
```js
|
|
assert.strictEqual(euler299(), 549936643);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function euler299() {
|
|
|
|
return true;
|
|
}
|
|
|
|
euler299();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|