We first prove that ∆BCA is a right triangle |

1 |
CP is congruent to CA |
They were both drawn with the same compass width |

2 |
PQ is congruent to AQ |
They were both drawn with the same compass width |

3 |
CQ is common to both triangles ∆PQC and ∆AQC |
Common side |

4 |
Triangles ∆PQC and ∆AQC are
congruent |
Three sides congruent (SSS). |

5 |
∠QCP, ∠QCA are congruent |
CPCTC. Corresponding parts of congruent triangles are congruent |

6 |
m∠QCA = 90° |
∠QCA and ∠QCP are a
linear pair and (so add to 180°)
and congruent so each must be 90° |

7 |
∆BCA is a right triangle |
∠BCA = 90°. |

We now prove the triangle is the right size |

8 |
CA is congruent to the given leg L1 |
CA copied from L1. See
Copying a segment. |

9 |
BC is congruent to the given leg L2 |
Drawn with same compass width |

10 |
∆BCA is a right triangle with the desired side lengths |
(7), (8), (9) |