Euclidian Tools

Def. 2.1 - A vector

A vector is defines from A to B[br]A vector is a line with direction, it requires a starting point and an ending point.

Figure 2.1: Example of Vectors

Figure 2.1 shows Vector AB, Vector CD, and Vector EF.

Based on Figure 2.1

Using Figure 2.1, manipulate points A through F. As you move each point, observe how vectors AB, CD, and EF change. Record your observations.

Def 2.2 The Circle

A circle can be drawn with any center and any radius.[br]The circumference is the total distance around the edge of a circle. It is the perimeter of a circle.[br]The radius is the distance from the center of the circle to the circumference.[br]A diameter is a straight line segment that passes through the center of a circle with both endpoints on its circumference.

Figure 2.2 - Circle Examples

Figure 2.2

Circle AB is defined by the radius AB with center at A[br]Circle CD is defined by the vector CD with center at C[br]Circle F is defined by the center point F

Question 2.2

Analyze the arrangement of circles as you manipulate the points. [br]Move points A through F and observe how the circles change.

Def. 2.3 Perpendicular line

A tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of tangency. The tangent of a circle is perpendicular to the radius at that point.

Figure 2.3.1 - Two Circles Share a Tangent

Figure 2.3.1 Perpendicular Lines

Circle AB is centered at A with radius AB. Circle BA is centered at B with radius BA. Circle CB is centered at C with radius CB. Circles AB, BA, and CB are all equal in size, they are said to be congruent.[br][br]Line DE passes through B and forms a tangent to circle AB and circle CB at point B.[br][br]Looking at angles ABD, DBE, EBC, and CBA, we can see they equally share the 2D plane. [br][br]By Euclid's definition, Line AC passing through B is perpendicular to Line DE passing through at point B.

Cut a line in half with a 90 degree angle

2.3.2 Perpendicular Bisector

Figure 2.3.2

AB is the radius of both circle AB (center A) and circle BA (center B). The two circles intersect at points C and D.[br][br]Line AC, AD, BC, and BD are all equal in length to AB (the radius of both circles).[br][br]Based on similar triangles:[br]Triangle ABC is congruent to triangle ADB by SSS.[br][br]AD = AC[br]DB = CB[br]AB = AB[br][br]Therefore, angle DAB = angle CAB. Since angle DAB + angle CAB = angle DAC,[br] then angle DAC = 2 * angle DAB.[br][br]Likewise, angle CBA = angle DBA. Since angle CBA + angle DBA = angle CBD,[br] then angle CBD = 2 * angle CBA.[br][br]We can say that line CD cuts line AB in half and E is the midpoint of AB.

Question 2.3.2

Complete the proof to show that CD is perpendicular to AB.[br][br](Hint: Use the congruent triangles AEC and AED, or BEC and BED, to show that angle AEC = angle AED = 90 degrees.)

Figure 2.3.3 - A circle and a chord

Figure 2.3.3: The diameter cuts the chord in half for a given circle.

Circle A has center A, with points B and C on its circumference.[br][br]Circle B and circle C intersect at point A (the center of circle A) and point D.[br][br]Circle B (center B) and circle C (center C) each have the same radius as circle A.[br][br]Line AD cuts chord CB at its midpoint.

Questions 2.3.3.1

Prove that AC, CD, DB, and BA are all equal in length.

Question 2.3.3.2

Prove that AD is the perpendicular bisector of chord CB.