Three Constructions and Their Proofs

Thomas Hobbes and the power of a proof:

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In his biography of the philosopher Thomas Hobbes, written around 1694, John Aubrey recorded the following. Hobbes was forty years old when he first encountered geometry, happening upon a copy of Euclid’s Elements lying open in a gentleman’s library. The book was open to Proposition 47 of Book I — the Pythagorean theorem.

‘He read the proposition. “By God,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps — and so on — that at last he was demonstratively convinced of that truth. This made him in love with geometry.’ Source: John Aubrey, Brief Lives, c. 1694.

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What Hobbes experienced is exactly what this lesson is about. He started from a conclusion that seemed impossible and worked backward through the chain of reasoning, step by step, until he reached premises he could not dispute. By the time he arrived at the axioms, he had no choice but to accept the original claim.

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The chain from Lesson 4.1 and Lesson 4.2 has the same structure. The triangle angle sum is true — but not because it is obvious. It is true because Proposition 31 (construct a parallel through a point) is true. And Proposition 31 is true because Proposition 23 (copy an angle) is true. And Proposition 23 is true because SSS is true. And SSS traces back to the axioms. Today we close that chain.

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This lesson does three things, and the third — the closure of the chain from Lesson 4.1 — is the most important. Begin class with the Hobbes story above. Do not rush it. Students who understand what Hobbes experienced understand what this lesson is about before the first task begins.

 

The angle copy construction (Task 1) often produces disappointment when groups find the method: it reduces to using Propositions 2 and 3 to copy lengths with a compass. That anticlimactic quality is worth naming honestly. The interesting mathematical work was establishing those propositions in the first place. The construction is their payoff. The proof that the construction works — the SSS argument — is where the real thinking lives.

 

Two student approaches from the first time this lesson was taught deserve to be preserved. When asked to construct a parallel line through a given point, one student built two equilateral triangles connected vertex to vertex, argued all four angles at the center intersection were right angles via SSS and CPCTC, then constructed a second equilateral triangle perpendicular to the original and used co-interior angles summing to 180° to establish parallel lines. A second student used the angle bisector (Proposition 9) to bisect a straight angle, then bisected that straight line and argued the resulting line was parallel to the first. Neither method produced the parallel through the given point exactly as described. But both arguments were mathematically sound and independently invented. Celebrate both before showing the standard method. A student who invents a valid argument for why something is true has done something more valuable than one who replicates a procedure.

 

The altitude construction (Task 3) is the most involved proof in the lesson. It chains every major tool from Unit 3 in sequence: SSS → CPCTC → SAS → CPCTC → linear pair. Students who work through it during the task will have experienced the satisfaction of a multi-step proof coming together from familiar pieces. Those who do not reach it will see in the notes that it is not magic — it is the same moves applied carefully in the right order.

 

Close the lesson by naming the full chain explicitly: Proposition 23 (angle copy) feeds into Proposition 31 (parallel through a point), which feeds into Proposition 32 (triangle angle sum). Everything from Lesson 4.1 is now fully proven from the axioms. Give this moment a full pause. Students have been carrying an open conditional since the previous lesson. Closing it is a genuine event.