This is exercise 11.2 in Geometry: Euclid and Beyond by Hartshorne.

Question: Two circles $\Gamma $ and $\Gamma' $that meet at a point A are tangent if and only if the tangent line to $\Gamma $ at A is equal to the tangent line to $\Gamma' $ at A.

Def'N: We say that a line $l$ is tangent to a circle $\Gamma $ if $l$ and $\Gamma $ meet in just one point A. We say that a circle $\Gamma $ is tangent to another circle $\triangle $ if $\Gamma $ and $\triangle $ have just one point in common.

We also have the following Corollary: If two circles meet at a point A but are not tangent, then they have exactly two points in common.

I also attached a list of the all axioms in the stated form that my book is using.

Incidence axioms

I1)For every two points A and B there exists a line a that contains them both.

I2)There exist at least two points on a line.

I3)There exist at least three points that do not lie on the same line.

Axioms of betweeness

B1) If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C.

B2) If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B.

B3) Of any three points situated on a line, there is no more than one which lies between the other two.

B4) Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.

Axioms of congruence

$C_1$. Given a line segment AB, and given a ray r originating at a point C, there exists a unique point D on the ray r such that $AB \cong CD$.

$C_2$. If $AB \cong CD$ and $AB \cong EF$, then $CD \cong EF$. Every line segment is congruent to itself.

$C_3$. (Addition). Given three points $A$, $B$, $C$ on a line satisfying $A * B * C$, and three further points D, E, F on a line satisfying $D * E * F$, if $AB \cong DE$ and $BC \cong EF$, then $AC \cong DF$.

C4. Given an angle $\angle BAC $ and given a ray $ DF$ there exists a unique ray $DE$ on a given side of the line $DF$ such that $\angle BAC \cong \angle EDF$

C5. Transitivity holds for congruence of angles and every angle is congruent to itself.

C6. (SAS)

E. (Circle-circle intersection Axiom).

Given two circles $\Gamma,\Gamma' $ if $\Gamma' $ contains at least one point inside $\Gamma $, and $\Gamma' $ contains at least one point outside $\Gamma$, then $\Gamma $and $\Gamma'$ will meet.

  • $\begingroup$ You are trying to prove this directly from axioms? No earlier proved theorems, such as "The tangent at $A$ is perpendicular to the diameter through $A$"? $\endgroup$ – Paul Sinclair Nov 11 '17 at 14:59
  • $\begingroup$ @PaulSinclair Just had a look, Proposition 11.2 of that book is precisely what you mentioned (it comes before the exercise). $\endgroup$ – N.Bach Nov 11 '17 at 16:25
  • $\begingroup$ Any proof using stuff derived from those axioms is fine, which includes your statement. $\endgroup$ – Faust Nov 12 '17 at 0:10

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