Find the locus of the vertex of a right angle whose two

Find the locus of the vertex of a right angle whose two sides are tangent to the cycloid $x=a(t-\sin t),\ y=a(1-\cos t)$.

Step 1: Tangent at the half-angle parameter $\tau=t/2$

Since $\dot x=a(1-\cos t)=2a\sin^2\tau$ and $\dot y=a\sin t=2a\sin\tau\cos\tau$, the slope is $y_x'=\cot\tau$. Writing the tangent through $P=(h,k)$ and simplifying: \[k\sin\tau - h\cos\tau = 2a\sin\tau - 2a\tau\cos\tau. \tag{1}\]

Step 2: Perpendicular tangents differ by $\pi/2$ in $\tau$

$\cot\tau_1\cot\tau_2=-1 \Leftrightarrow \cos(\tau_1-\tau_2)=0 \Leftrightarrow \tau_2=\tau_1-\pi/2$.

Step 3: The two tangency conditions

Using (1) at $\tau$ and at $\tau-\pi/2$ (with $\sin(\tau-\pi/2)=-\cos\tau,\ \cos(\tau-\pi/2)=\sin\tau$): \[(k-2a)\sin\tau + (2a\tau-h)\cos\tau = 0, \qquad \bigl(2a\tau-a\pi-h\bigr)\sin\tau + (2a-k)\cos\tau = 0. \tag{2}\]

Step 4: Solve the homogeneous pair

Let $u=h-2a\tau$. From the first equation, $\cot\tau=(k-2a)/u$; substituting into the second (where $2a\tau-a\pi-h=-(u+a\pi)$) and multiplying through by $\sqrt{u^2+(k-2a)^2}$:

\[ u^2 + a\pi\,u + (k-2a)^2 = 0. \tag{3} \]

Step 5: Extract $h$ and $k$ as functions of $\tau$

From (3), $u^2+(k-2a)^2=-a\pi u$; combined with $\sin^2\tau=u^2/\bigl[u^2+(k-2a)^2\bigr]$ this gives $u=-a\pi\sin^2\tau$, i.e.\ $h=2a\tau-a\pi\sin^2\tau$. And $(k-2a)^2=-u(u+a\pi)=a^2\pi^2\sin^2\tau\cos^2\tau$, so $k-2a=\pm a\pi\sin\tau\cos\tau$; tracking the same branch used for $u$ (the one consistent with (3) rather than treating the square root independently) selects the minus sign:

\[ h = 2a\tau - \frac{a\pi}{2}(1-\cos2\tau), \qquad k = 2a - \frac{a\pi}{2}\sin2\tau. \tag{4} \]

Step 6: Verification

\[a=1.4,\ \tau=0.4,1.0,1.9,2.6,-0.7\text{: the numerically computed intersection }(h,k)\text{ of the tangents at }\tau,\tau-\pi/2\text{ matches }(x(\varphi),y(\varphi))\text{ at }\varphi=2\tau\text{ exactly}\ \checkmark\]

\[\text{Cross-check: pairing }(\tau,\tau+\pi/2)\text{ instead reproduces points already on the same curve, at a shifted }\varphi\approx3.9416\text{ rather than a second, separate branch}\ \checkmark\]