1 |
Line segments AP, AQ, PB, QB are all congruent |
The four distances were all drawn with the same compass width c. |

Next we prove that the top and bottom triangles are
isosceles and
congruent |

2 |
Triangles ∆APQ and ∆BPQ are
isosceles |
Two sides are congruent (length c) |

3 |
Angles AQJ, APJ are congruent |
Base angles of
isosceles
triangles are congruent |

4 |
Triangles ∆APQ and ∆BPQ are
congruent |
Three sides congruent (sss). PQ is common to both. |

5 |
Angles APJ, BPJ, AQJ, BQJ are congruent.
(The four angles at P and Q with red dots) |
CPCTC. Corresponding parts of congruent triangles are congruent |

Then we prove that the left and right triangles are
isosceles and congruent |

6 |
∆APB and ∆AQB are
isosceles |
Two sides are congruent (length c) |

7 |
Angles QAJ, QBJ are congruent. |
Base angles of
isosceles
triangles are congruent |

8 |
Triangles ∆APB and ∆AQB are
congruent |
Three sides congruent (sss). AB is common to both. |

9 |
Angles PAJ, PBJ, QAJ, QBJ are congruent.
(The four angles at A and B with blue dots) |
CPCTC. Corresponding parts of congruent triangles are congruent |

Then we prove that the four small triangles are congruent and finish the proof |

10 |
Triangles ∆APJ, ∆BPJ, ∆AQJ, ∆BQJ are
congruent |
Two angles and included side (ASA) |

11 |
The four angles at J - AJP, AJQ, BJP, BJQ are
congruent |
CPCTC. Corresponding parts of congruent triangles are congruent |

12 |
Each of the four angles at J are 90°. Therefore AB is perpendicular to PQ |
They are equal in measure and add to 360° |

13 |
Line segments PJ and QJ are congruent.
Therefore AB bisects PQ. |
From (8), CPCTC. Corresponding parts of congruent triangles are congruent |