Chapter 6

As Stella entered the forest, she realized just how correct Pythagoras had proved to be. The forest was blacker than night. She became very grateful of the little lantern Pythagoras had given her. Surprisingly, it emanated no heat. She wondered if Geomno had more advanced inventions and contraptions than her home. Stella caught a glimpse of an odd looking figure down the road, and quickened her pace so that she might see it. The object turned out to be another huge pink triangle, but this one was not a prism. It was only 2-D and looked as if it were a road sign, but there weren't any turns or forks in the road. Stella looked more closely at it and observed that it had writing. She read the words out loud.

"A segment bisector intersects a segment at its midpoint. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. A point is equidistant from two points if its distance from each point is the same.

Perpendicular bisector theorem:" Stella groaned. Not another theorem! "…If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. When a point is in the same distance from one line as it is from another line, then the point is equidistance from the two lines.

Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle." How interesting, thought Stella

"A perpendicular bisector of a triangle is a line that is perpendicular to a side of the triangle at the midpoint of the side. When three or more lines intersect in the same point, they are called concurrent lines. The point of intersection of the lines is called the point of concurrency. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

Concurrency of Perpendicular Bisectors of a Triangle: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. It is always inside the triangle.

Concurrency of Medians of a Triangle: The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. It can lie inside, on, or outside the triangle. Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle.

Concurrency of Altitudes of a Triangle: The lines containing the altitudes of a triangle are concurrent.

A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long" Stella sighed. That was quite a bit to take in! She caught a few extra words at the base of the triangle which she hadn't bothered in reading before. "I hope you find this helpful, Stella. Pythagoras." Of course, Stella thought. Pythagoras did, after all, live in a triangle house. She took one last glance at the triangle before starting off again, musing over the facts she had just learned.

After many minutes of walking (perhaps hours), Stella became impatient. Where was Pythagoras' friend? Had she missed a turn? Stella become worried, until suddenly, right in front of her, she glimpsed a gate. Hurriedly, Stella crossed over to it.

The gate was constructed of thick oaken doors. It was very tall, and was wide enough to span the entire road. The odd thing about it, however, was the writing engraved in the doors. Stella read the intricately carved words out loud to herself.

"All read, ye who wish to enter here…A polygon is a plane figure. It must be formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. Each side must intersect exactly two other sides, one at each endpoint. Each endpoint of a side is a vertex of the polygon. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is called nonconvex, or concave. A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is equiangular and equilateral. A diagonal of a polygon is a segment that joins two nonconsecutive verticals.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. The nonparallel sides are the legs of the trapezoid. It the legs of the trapezoid are congruent, then the trapezoid is an isosceles trapezoid." Stella finished reading.Experimentally, she pushed at the heavy wooden door. Stella was thoroughly surprised when it proved to open without so much as a squeak. Beyond the gate, the pathway led downhill. Stella could not see much beyond the warm glow of the lamplight, and was hesitant to follow the path. Oh well, she thought. This road hasn't let me down yet; I've never encountered danger. And with a shrug of her shoulders, she set off downwards.

As she rounded one particularly sharp corner, her eye caught on something. A flash of silver had shone in the darkness of the trees. Stella furtively hoped that it wasn't the eyes of some wild beast that owned that silver flash. In the distance, she began to detect a slight gurgling sound, such as one would find near a stream. Stella squinted into the darkness. It may have been her imagination, but Stella saw a speck of light in the distance, that didn't have the same warm glow of her trusty lantern.

As she drew nearer, Stella became almost positive that there was a light up ahead. She glanced at it every once in a while as she walked, trying to see what might be lurking in the light. Finally, she became near enough to see.

The source of light was a floating, glowing contraption that hung over a small pool. A small trickle of a waterfall fed the pool, and that was where Stella had detected the gurgling noise. A large wooden podium that Stella distinctly remembered from somewhere in her past sat near the lake, also bathed by the pool of light.

Stella gasped and ran towards the podium. Maybe, if I could only hope…The Geomno might be there…, she thought as she ran. Sure enough, the leather-bound book sat on top of the podium, looking exactly as Stella had remembered it from her first encounter. Slowly, she flipped open the cover.

The wind picked up at once, tugging at Stella's dress and hair. Moments later a black window grew from the book, and pulled Stella into its dark vortex…

The End