Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex. (Hint: the three lines divide the plane into 7 regions.)
Step-by-step solution
Idea: In ABCD the sides are AB, BC, CD, DA, and the diagonals are AC and BD. ABCD is convex exactly when the diagonals AC and BD cross, so D must be on the other side of line AC from B. A dent appears when one vertex lies inside the triangle of the other three. Sides cross when D is just across AB or just across BC.
- The 7 regions: the inside of ∆ABC; three regions each sharing a side with the triangle (across AB, across BC, across CA); and three corner regions, each touching the triangle only at a vertex.½ mark
- D inside ∆ABC: D is inside the triangle of the other three vertices, so the internal angle at D is reflex. ABCD is non-convex.½ mark
- D across CA (on the other side of line CA from B, but on the same side of AB as C and of BC as A): the diagonals AC and BD cross. ABCD is convex.1 mark
- D across AB only: segment CD crosses side AB, so ABCD is self-intersecting. D across BC only: segment DA crosses side BC, so ABCD is again self-intersecting.1 mark
- D in a corner region: beyond B, the vertex B lies inside ∆ACD; beyond A, A lies inside ∆BCD; beyond C, C lies inside ∆ABD. In each case one internal angle is reflex and no sides cross: non-convex.1 mark
Check: With A(0, 0), B(6, 0), C(2, 4): D(1, 1) inside → non-convex; D(0, 3) across CA → convex; D(3, −2) across AB → self-intersecting; D(5, 3) across BC → self-intersecting; D(−2, −1), D(8, −1), D(2.2, 6) in the corner regions → non-convex. (Checked by computer for 200 000 random positions: each region always gives the same type.)
Answer to write in the exam
Lines AB, BC, CA make 7 regions: inside ∆ABC, 3 regions across a side, 3 corner regions
D inside ∆ABC ⇒ reflex ∠D ⇒ non-convex
D across CA (opposite side of CA from B) ⇒ diagonals AC, BD intersect ⇒ convex
D across AB only ⇒ CD crosses AB ⇒ self-intersecting; D across BC only ⇒ DA crosses BC ⇒ self-intersecting
∴ D in a corner region (beyond A, B or C) ⇒ that vertex lies inside the triangle of the other three ⇒ non-convex
Common mistakes that cost marks
- Treating all three “across a side” regions alike. Which side matters: across CA (the diagonal side) gives convex, across AB or BC gives crossing sides.
- Forgetting the three corner regions, so counting only 4 regions.
- Joining D to B instead of to C and A. In ABCD, D is joined to C and to A.
How this can come in the exam
A(0, 0), B(4, 0), C(0, 4). For which position of D is ABCD convex?
- D(1, 1)
- D(−2, 2)
- D(2, −2)
- D(3, 3)
Show answer
(B) D(−2, 2)
For a convex ABCD, D must be on the other side of line CA (the y-axis) from B, while staying on C’s side of AB and on A’s side of BC. Only D(−2, 2) does this. D(1, 1) is inside the triangle (non-convex); D(2, −2) and D(3, 3) make two sides cross.
Try one yourself
A(0, 0), B(6, 0), C(2, 4). Classify ABCD for D(2, 1) and for D(−1, 3).
Show answer
D(2, 1) is inside ∆ABC: non-convex. D(−1, 3) is across line CA from B: convex.
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