Euclidian geometry: Construct circle trough point on angle bisector where

In summary, the conversation discusses how to construct a circle using a given angle with its bisector and a point on that bisector. The solution involves finding a point M and B such that AM = BM, and constructing a perpendicular at A to intersect with BO at T. The formed angle BTA acts as a guide for constructing the desired circle.
  • #1
Berrius
19
0

Homework Statement


This is part from a larger construction, but I realized if i can construct this, i can do the larger construction. All ofcourse with ruler and compass.
I have been given an angle with its bisector and a point on that bisector. I have to construct a circle trough that point (A), such that the lines from the angle are tangent lines to the circle.

The Attempt at a Solution


attachment.php?attachmentid=40291&stc=1&d=1319461727.jpg

I have to find a point M and B such that AM = BM, and than i can construct the circle. But i have no idea how to do this.
 

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  • #2
Let's label the vertex to the right, O.
Construct a perpendicular at A, and extend it out to meet BO at T.
We have located two tangents to the circle, one being BO and the second being TA. These tangents together form angle BTA.

Hint: this angle forms a sort-of-hat which symmetrically sits atop the desired circle.
 

1. What is Euclidean geometry?

Euclidean geometry is a branch of mathematics that deals with the study of shapes, sizes, and positions of objects in space. It is based on the work of the ancient Greek mathematician Euclid and is the most widely used system of geometry in modern mathematics.

2. What is an angle bisector in Euclidean geometry?

An angle bisector is a line or ray that divides an angle into two equal parts. It is a fundamental concept in Euclidean geometry and is often used to solve geometric problems involving angles.

3. How do you construct a circle through a point on an angle bisector?

To construct a circle through a point on an angle bisector, you will need to use a compass and a straightedge. First, draw the angle bisector using a straightedge. Then, place the compass on the point and draw arcs that intersect the angle bisector on either side. Finally, use the compass to draw a circle that passes through the point and intersects the two arcs.

4. What are some real-world applications of Euclidean geometry?

Euclidean geometry has numerous real-world applications, including architecture, engineering, navigation, and design. It is also used in computer graphics, computer-aided design, and 3D modeling.

5. What are some key principles of Euclidean geometry?

Some key principles of Euclidean geometry include the concepts of points, lines, and planes, as well as the properties of shapes such as triangles, circles, and polygons. It also includes the rules of congruence, similarity, and symmetry, as well as the Pythagorean theorem and the laws of geometric transformations.

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