Find the Equilateral Triangle Trivia Quiz

Answer: Finite

Euclid's Proposition I.1 uses as its starting point a finite straight line, which is to say a line with a start point and an end point. It is also referred to in other (more modern) texts as a straight line segment. In order to create this finite straight line in the first place, he relies on his first postulate - in essence, that a finite straight line can be drawn between two points (A and B) - alongside a number of his definitions for terms such as "line" and "point".

Proposition I.1 says that an equilateral triangle can be constructed on such a finite straight line.

Answer: Radius

The first significant development in Proposition I.1 is to use the finite straight line (coloured red) as the basis for the creation of a circle (Point A at centre and Point B on the circumference) as set out in Postulate 3. The finite straight line is now also the radius of this circle. This circle could be drawn using a compass, with the needle point placed at A and marker point placed initially at B (then rotated 360 degrees).

Whilst Euclid used "diameter" in 'Elements', "radius" is absent. It is instead referred to as a "distance" or "straight line from the centre". Euclid wrote 'Elements' in Greek and "radius" is from Latin; this helps to largely explain why the specific word "radius" was not used, but doesn't explain why the relevant concept wasn't provided a different label.

Answer: Yes

Postulate 3 is used for a second time, this time creating a circle with Point B at the centre and Point A on its circumference. This creates a situation where the original finite straight line (in red) is the radius of both circles. In Euclidean terms, a circle is defined by having a centre point and the distance of the finite straight line from the centre point to the boundary (circumference). As the "distance" is the same in both circles, then the circles are the same size.

This is easier to see algebraically. If the grey circle with its centre at A has radius, R1, and the black circle with its centre at B has radius, R2, then using the equation for the area of a circle:

Area of grey circle, A(g) = π * (R1)^2
Area of black circle, A(b) = π * (R2)^2

The radii are the same (the red line), therefore R1 = R2. By replacing R1 with R2 in the equation for the area of the grey circle:

A(g) = π * (R2)^2
A(b) = π * (R2)^2

These two expressions are the same, therefore A(g) = A(b), and we can say the two circles have the same area.

Answer: Point of Intersection

At this point in Proposition I.1, Euclid outlines that at point C, "the circles cut one another"; a more modern formulation of this description is that the two circles intersect at point C or that C is a point of intersection of the two circles. The diagram also shows that there is a second point of intersection (below the red line).

Whilst this reasoning feels intuitively correct, there have been objections raised in relation to the idea that these two circles - must - have a point of intersection at C. The first objection is that, at the very least, the formulation that two such circles "cut one another" in such circumstances should be listed as a "self evident" postulate. The more fundamental objection however relates to the fact that Euclid makes a number of assumptions about the nature of the plane on which the circles lay. Analysis of this particular assumption is fairly lengthy and deals with the differences between the likes of the real and rational planes.

Answer: AC = AB

In modern terms, we can simply say that both the red line (AB) and the blue line (AC) are radii of the circle which has its centre point at A. All radii of the same circle are of the same length (from Definition 15) and so the red line is equal in length to the blue line. This can be expressed as AC = AB.

Answer: BC = AB

The approach taken to the construction of a finite straight line between points A and C (blue line) is replicated at this stage in order to create another finite straight line extending from point B to point C (yellow line). As both the red line and the yellow line are radii of the circle with the centre at point B, then the yellow line (BC) is equal in length to the red line (AB).

More succinctly, BC = AB.

Answer: Things which are equal to the same thing are also equal to one another

This common notion is a rather wordy way of saying if A = B and C = B, then A = C. Applied to Proposition I.1, it allows Euclid to reason that because line AC is equal in length to line AB and that line BC is equal in length to line AB, then line AC is equal in length to line BC. Given that the lengths of all three lines of the trilateral structure ABC are the same, then the structure ABC is said to be an equilateral triangle (Definition 20, "an equilateral triangle is that which has its three sides equal").

And so it has been shown - notwithstanding a number of challenges to the logic over the millennia - that an equilateral triangle can be constructed on a finite straight line.

Answer: 60 degrees

One of the striking features of the Euclidean method is that he put forward a logical approach to constructing geometric objects through the use of simple straight lines and circles; the lack of numbers and algebra is in complete contrast to the geometry of today. Whilst Proposition I.1 does not reference the value of the angles in an equilateral triangle, it would be one of the first things a student of triangles would learn today. All interior angles of an equilateral triangle are equal to each other; they are all sixty degrees.

I still find it remarkable that the propositions set out in 'Elements' were devised over 2300 years ago!

Answer: 120 degrees

Euclid's second postulate from Book I of 'Elements' allows for a finite straight line to be extended indefinitely. In this case the straight line is extended to point D rather than to infinity and in so doing a new finite straight line from A to D is created. The interior angles of an equilateral triangle are all sixty degrees and angles along a straight line sum to 180 degrees.

The angle between lines BD and BC is therefore 180 - 60 = 120 degrees.

Answer: Reflex

As the triangle is equilateral, each of the interior angles is equal to sixty degrees. One way to describe angle "x" would be that, when added to the sixty degree angle between line AC and line BC, "x" completes a circle. A circle has 360 degrees and so x = 360 - 60 = 300 degrees.

By definition, any angle that has a value between 180 degrees and 360 degrees (not inclusive) is a reflex angle. An acute angle is one that takes a value in the range of zero to ninety degrees (not inclusive). Whilst Euclid did provide a definition for an "acute angle" in 'Elements', he did not make reference to what is now referred to as a "reflex angle".