Proof that a secant line intersects a circle in exactly two points (according to Hilbert's axiomatic system)

Question

With Hilbert's axiomatic system, How do I prove that a non-tangent line $d$ that intersects a circle $C$ intersects it in exactly two point?

My teacher gave us the following clue: First show that if ABC and A'B'C' are two triangles with right angles in B and B' and if AB≅A'B' and AC≅A'C' then the triangles are congruent.

Do you want an analytic proof, that if $y=mx+b$ intersects $x^2+y^2=1$, then there are either one or two points of intersection? — Akiva Weinberger, Nov 16 '15 at 14:57

score 1 · Answer 1 · answered Nov 16 '15 at 15:26

1

Suppose $AC$ is the diameter of the circle, and $\ell$ is the line. Suppose that the point $B$ is an intersection of the line and the circle. We know that $ABC$ is a right triangle with a right angle at $B$. You want to find a point $B'$ such that $AB'C$ is a right triangle and such that $B'$ lies on $\ell$.

answered Nov 16 '15 at 15:26

Akiva Weinberger

23,062

Can I just say that in any circle with a diameter AC we can build two right angled triangles ABC and AB'C so that B and B' $\in$ $l$ – Mathphilo Nov 16 '15 at 15:39

score 0 · Answer 2 · answered Nov 16 '15 at 14:55

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whats there to prove draw a circle draw a chord . the chord by definition joins any two points of a circle e g. diameter (longest chord) so you have proved by definition that secant(chord) cuts circle in exactly two points.

answered Nov 16 '15 at 14:55

Archis Welankar

15,910

my teacher gave us the following clue: show that if ABC and A'B'C' are two triangles with right angles in B and B' and if AB≅A'B' and AC≅A'C' then the triangles are congruent – Mathphilo Nov 16 '15 at 14:59
so how is their congruency related to a circles secant. – Archis Welankar Nov 16 '15 at 15:05
I really have zero clue – Mathphilo Nov 16 '15 at 15:06

Proof that a secant line intersects a circle in exactly two points (according to Hilbert's axiomatic system)

2 Answers2