Construction of Circumcircle and Incircle of a Triangle

A locus is the set of all points, and only those points, whose coordinates satisfy one or more specified geometric conditions:

• Locus 1 (Perpendicular Bisector): The locus of points equidistant from two fixed points $A$ and $B$ is the perpendicular bisector of segment $AB$. Any point $P$ lying on this bisector satisfies $PA = PB$.
• Locus 2 (Angle Bisector): The locus of points equidistant from two intersecting rays $BA$ and $BC$ is the internal angle bisector of $\angle ABC$. Any point $Q$ lying on this ray is equidistant from lines $BA$ and $BC$.

Follow this rigorous sequence of operations to construct the circumcircle of triangle $ABC$:

1. Step 1 (Construct the Triangle): Using straightedge and compass, draw $\triangle ABC$ according to the given measurements of sides and/or angles.
2. Step 2 (First Perpendicular Bisector):
   Place the compass point at vertex $B$. Set the compass opening to more than half the length of side $BC$ (radius $> \frac{1}{2} BC$).
   Draw two arcs, one above and one below side $BC$.
   Without changing the compass opening, place the compass point at vertex $C$ and draw two arcs intersecting the previous arcs at points $P$ and $Q$.
   Join $PQ$ with a straight line. Line $PQ$ is the perpendicular bisector of side $BC$.
3. Step 3 (Second Perpendicular Bisector):
   Similarly, with opening $> \frac{1}{2} AB$, draw arcs from vertices $A$ and $B$ to construct the perpendicular bisector of side $AB$.
   Draw this bisector line to intersect the first bisector $PQ$ at a single point $S$.
4. Step 4 (Locate Circumcenter S & Measure Circumradius R):
   Point $S$ is the circumcenter.
   Join $S$ to vertex $A$ (or $B$ or $C$) with a dashed line segment.
   Measure the length $R = SA$. By theorem, $SA = SB = SC = R$.
5. Step 5 (Draw the Circumcircle):
   Place the needle point of the compass at center $S$, open the pencil point to vertex $A$, and draw a continuous, unbroken circle.
   Verify visually that the circumference passes smoothly and cleanly through all three vertices $A, B$, and $C$.

A high-yield conceptual question in Madhyamik examinations asks students to identify or verify the position of the circumcenter:

• Acute-Angled Triangle: All angles are $< 90^\circ$. The circumcenter $S$ lies strictly inside the triangle.
• Right-Angled Triangle: One angle is $90^\circ$. The circumcenter $S$ lies exactly on the midpoint of the hypotenuse!
  Crucial Property: In a right triangle with hypotenuse $c$, the circumradius is exactly half the hypotenuse: $$R = \frac{\text{Hypotenuse}}{2}$$ Vertex opposite the hypotenuse is inscribed in a semicircle ($90^\circ$ by Thales' theorem).
• Obtuse-Angled Triangle: One angle is $> 90^\circ$. The circumcenter $S$ lies strictly outside the triangle, situated opposite the obtuse-angled vertex.

Follow this exact sequence of operations to construct the incircle of triangle $ABC$:

1. Step 1 (Construct the Triangle): Draw $\triangle ABC$ using the given dimensions.
2. Step 2 (First Angle Bisector):
   Place the compass tip at vertex $B$. With any convenient radius, draw an arc cutting side $BC$ at point $X$ and side $BA$ at point $Y$.
   With centers $X$ and $Y$ and a radius $> \frac{1}{2} XY$, draw two arcs in the interior of $\angle B$ that intersect at point $M$.
   Draw ray $BM$. Ray $BM$ is the internal bisector of $\angle B$.
3. Step 3 (Second Angle Bisector):
   Similarly, place the compass at vertex $C$ and construct the internal bisector of $\angle C$, drawing ray $CN$.
   Ray $BM$ and ray $CN$ intersect at a unique point $I$.
   Point $I$ is the incenter of the triangle.
4. Step 4 (Crucial Step: Drop Perpendicular from Incenter to a Side):
   Do NOT simply measure a random distance to a side!
   Place the compass needle at incenter $I$. Open the compass slightly larger than the distance to base $BC$.
   Draw an arc cutting side $BC$ at two distinct points $E$ and $F$.
   From points $E$ and $F$, with radius $> \frac{1}{2} EF$, draw two intersecting arcs below $BC$ meeting at point $K$.
   Join $IK$ with a straight line intersecting $BC$ at point $D$.
   Line segment $ID$ is strictly perpendicular to side $BC$ ($ID \perp BC$).
   The length of $ID$ is the exact inradius $r$.
5. Step 5 (Draw the Incircle):
   With needle at incenter $I$ and radius equal to $ID$, draw a circle.
   The circle will touch side $BC$ at point $D$, side $AB$ at a point, and side $AC$ at a point internally.

The inradius $r$ is analytically related to the area $\Delta$ and semi-perimeter $s$ of the triangle by: $$\Delta = \text{Area}(\triangle IBC) + \text{Area}(\triangle ICA) + \text{Area}(\triangle IAB)$$ $$\Delta = \frac{1}{2} a r + \frac{1}{2} b r + \frac{1}{2} c r = r \left(\frac{a + b + c}{2}\right) = r \cdot s$$ $$\mathbf{r = \frac{\Delta}{s}}$$ For a right-angled triangle with legs $a, b$ and hypotenuse $c$: $$\mathbf{r = \frac{a + b - c}{2}}$$ Students can use this formula to pre-calculate and verify their compass measurement of $r$!

In Madhyamik examinations, Section IV (Constructions / সম্পাদ্য) carries a mandatory 5-mark question. Examiners grade according to strict mechanical drafting criteria:

• Pencil Hardness: Use a finely sharpened 2H pencil for construction arcs and lines, and an HB pencil for the final circle and triangle outline. Thick, blunted lines lead to a $\pm 2\text{ mm}$ error, causing automatic mark deduction.
• No Multiple Over-Tracing: Draw every arc and circle in a single smooth stroke. Overwritten, hairy, or double lines result in a 1-mark deduction.
• Retain Construction Arcs: Never erase your construction arcs! The intersecting compass arcs are the proof of your geometric method. If arcs are erased, examiners cannot verify the construction and deduct 2 marks.
• Explicit Measurement Sentence: Always conclude your construction with an explicit written statement:
  "The circumradius is measured as $R = 3.8\text{ cm}$ (or inradius $r = 1.9\text{ cm}$)". An error within $\pm 0.1\text{ cm}$ is accepted by board examiners without penalty.