Kayla Proof 3.22d,g
- Author:
- Kayla Moore
Refer to the construction above which is outlined as Exercise 3.22 in College Geometry.
ABCD
d. Let AEC and BEC be triangles. Since line CD was created using Euclid's Postulates and Proposition 10, we know that CD bisects line AB because the circles used to create line CD are equivalent in size because their radii were equal. Since line CD bisects line AB, we know that line AE is equal to line EB. Also notice CE is shared by both triangles, so this side length is also equal. From Proposition 8 we know that if two triangles share two equal side lengths and their bases are equal, we know that the angle between the two equal side lengths are also equal. This means that angle AEC is equal to angle BEC. Proposition 13 tells us that if a straight line stands on a straight line, as CE is doing to AB, then the sum of the two angles is equivalent to two right angles or 180 degrees. Based on these two facts, we can see that:
because
Since angle AEC is equivalent to angle BEC, they both are equal to 90 degrees. Based on the definition of a right angle, both angle AEC and angle BEC are 90 degrees. So by the definition of perpendicular as defined by Definition 10, line CE cuts line AB perpendicularly.
ACBD is a rhombus.
g. Let ACBD be a quadrilateral. Let's look at the four triangles in the quadrilateral. Recall from the previous discussion that since Proposition 10 and Euclid's Postulates were used to create the line CD, we know that line AE and line EB are equivalent in length. This also means that line CE is equivalent in length to line CE and line ED. From above, we also know that angle AEC and angle BEC are equal in measure. Similarly, angle AED is equivalent in measure to angle BED. By Proposition 8, recall, that triangles are congruent if two side lengths are equal and the middle angles are also equal. Since we know that line AE is equal to line EB and line EC is equal to line EC by Common Notion 1 and we previously showed that angle AEC and angle BEC are also equal, so the lines AC and BC must also be equal. Similarly, lines AD and BD are equal in length. Since the lines that create the four triangles are equivalent respectively, the four lines that create the quadrilateral must also be equivalent in length.
Look at triangle AEC. Recall that the sum of all of the angles in a triangle is 180 degrees. From the previous proof that we know that the angle AEC is a right angle so it is 90 degrees. From these two facts, we know that the sum of the angles EAC and ECA would have to be equal to 90 degrees. From Proposition 18 and 19, we know that the angles opposite side lengths are respectively smaller or larger than other side lengths and angles. Since the triangle AEC was created by the same process that would create an equilateral triangle as described in Proposition 1. Since the line CD bisects the equilateral triangle, we know that the side length of line AE and line CE would be different. Since the lines are different lengths, we know that the angles opposite these side lengths must also be different for previously described reasons. Since the angles are different in measure but their sum must equal 90 degrees, we know that neither of the angles could be 45 degrees. This is true for all of the triangles CEB, BED and AED for the same reasons. Since the quadrilateral in question is composed of these triangles, we know that angles of the quadrilateral would not be right angles.
By Definition 22, we know that a rhombus must be equilateral and not right-angled. From the arguments above, we know that the quadrilateral ACBD is a rhombus.