SOLUTION: Prove the following theorem: There exists one circumcircle for any triangle, or any triangle is cyclic.

Question 1195525: Prove the following theorem: There exists one circumcircle for any triangle, or any triangle is cyclic.
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52866)   (Show Source): You can put this solution on YOUR website!
.

Usually and traditionally in Geometry, this statement is formulated in this EQUIVALENT form

        Three perpendicular bisectors of a triangle sides are concurrent, in other words, they intersect at one point.
        This intersection point is equidistant from the three triangle vertices and is the center of the circumscribed circle of the triangle.

For the proof, see the lesson
    - Perpendicular bisectors of a triangle sides are concurrent
in this site.

/////////////////////

At this site, you have free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lesson is the part of this online textbook under the topic "Properties of triangles".

Save the link to this online textbook together with its description

Free of charge online textbook in GEOMETRY
https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson

to your archive and use it when it is needed.

        This textbook contains whole/(entire) standard high-school
        Geometry curriculum in its logical development from " a " to " Z ".

        Euclid retold online in English by me and in my own words,
                as I remember it from my high school years.

Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Consider any 3 non-collinear points ,, .
The center of the circle passing through any two given points lies on the perpendicular bisector of the line segment joining those points. This is easy to prove.

Let be the center of any circle passing through the points , .
Join , and draw be perpendicular to , touching at .

In Δ s , :
(radius)
is common
∡ =∡ ∘ (By construction).
So, Δ =Δ .
Therefore ⟹ ⊥ bisector of .
Now consider the perpendicular bisectors of line segments and .
As the points , and are not collinear, the perpendicular bisectors of and are not parallel, and must intersect; two lines can intersect at only one point (let’s call it ), which must then be the of the passing through the 3 given points (trivial to show that ).

As the points , and are non-collinear, they could be the vertices of a . Hence the proof.

RELATED QUESTIONS
Could someone outline a strategy for solving the following problem? *Let \( \triangle... (answered by CPhill)
Need help answering please? Study the flow proof for a leg-leg (LL) Congruence... (answered by MathLover1)
Prove that for any prime k>2, some prime k-2 or k+2... (answered by ikleyn)
Hi am slightly stuck on this question in geometry Given a triangle ABC prove that:... (answered by richard1234)
Prove lines containing the altitudes of any triangle are... (answered by richard1234)
The bisector of a vertical angle A of a triangle ABC meets the circumcircle of triangle... (answered by venugopalramana)
I need help with the following proof and CANNOT use any givens like all the angles of... (answered by richard1234)
Through any ____ points there exists exactly one... (answered by Alan3354)
Fundamental Ideas Points, Lines, and Planes Postulates and Theorems Segments,... (answered by richard1234)